Laurent Desbat
email: laurent.desbat@univ-grenoble-alpes.fr.

Rolf Clackdoyle

‡ L Desbat

Calibration and data consistency in parallel and fan-beam linogram geometries

Calibration and data consistency in parallel and fan-beam linogram geometries

Laurent Desbat, Rolf Clackdoyle

Introduction

There continues to be increasing interest in geometric calibration procedures for computed tomography, because small misalignments of the projections can dramatically degrade the quality of reconstructed images. Self-calibration, whereby unknown geometric misalignments are corrected concurrently with the production scan is also becoming a popular approach ( [START_REF] Aichert | Epipolar consistency in transmission imaging[END_REF][START_REF] Lesaint | Calibration for circular cone-beam ct based on consistency conditions[END_REF]) although there are still challenges to be overcome.

Self-calibration methods generally appeal to data consistency conditions, because geometric misalignments can break down some kinds of consistency [START_REF] Basu | Uniqueness of tomography with unknown view angles[END_REF][START_REF] Basu | Feasibility of tomography with unknown view angles[END_REF]. For example, it is known that the order-1 and order-0 Helgason-Ludwig consistency conditions [START_REF] Ludwig | The radon transform on euclidean space[END_REF][START_REF] Helgason | The Radon Transform[END_REF][START_REF] Natterer | The Mathematics of Computerized Tomography[END_REF] provide information on the center of mass of the object being imaged. Misaligned projections would then reveal inconsistent information on the center of mass and could possibly provide a criterion for correcting these misalignments. On the other hand, order-0 conditions alone only provide the total mass of the object and would not be useful for detecting misalignments. Several other well-known facts (which are all reviewed below) are that if the projection angle is reversed, a mirror version of the reconstructed image occurs; that a fixed shift in the projection angle will just rotate the reconstructed image by the same angle, but that a fixed shift in the ray variable produces inconsistent projection data.

Although the most common (2D) tomographic imaging configuration uses parallel or fan-beam projections measured on a circle, linogram geometries, whereby a flat detector is placed opposite fan-beam sources following a linear path are also relevant, particularly for non-medical applications such as non-destructive testing, baggage security scanning, etc.

Here, we present the corresponding data consistency properties for the parallel and fan-beam linogram geometries. We explore the center of mass and total mass considerations, we examine the effect of a constant shift of the projection index or of the ray index, and we consider reversing the projection index. These results provide the corresponding information for linogram self-calibration issues as is known for the conventional circular tomographic model.

The 2D Radon transform corresponds to the circular tomographic model. We begin by expressing the stated (and well-known) results mathematically for this case. Then we take the small step to parallel linogram projections and even there we see some interesting differences (e.g., the rotation of the object becomes a linear shear transformation). Finally we examine the fan-beam linogram projections. These linogram results, derived from known linogram consistency conditions, are new.

2D Radon transform

We let µ ∈ L 1 (R 2 ) represent the 2D object function. We define parallel projections by

p φ (s) . . = p(φ, s) . . = Rµ(φ, s) . . = R µ(s θ φ + l ζ φ )dl (1)
where φ ∈ [0, 2π), s ∈ R, θ φ = (cos φ, sin φ) , ζ φ = (-sin φ, cos φ), so the parallel projection at angle φ is p φ . See Fig. 1. We will always use the first variable to indicate the projection index. The Helgason-Ludwig consistency conditions [START_REF] Natterer | The Mathematics of Computerized Tomography[END_REF] are expressed in terms of moments of the parallel projections.

Π n (φ) = ∀n ∈ N, R s n p(φ, s)ds (2)
As is well-known, the function p is consistent (meaning that there exists some µ such that p = Rµ) if and only if, ∀n = 0, 1, 2, . . . , Π n is a homogeneous polynomial in cos φ and sin φ of degree n (or else the zero polynomial). I.e. Π n is of the form

Π n (φ) = n k=0 a k cos n-k (φ) sin k (φ) (3)
which can be alternatively expressed as

Π n (φ) =A n cos n (φ -A n-1 ) + A n-2 cos n-2 (φ -A n-3 ) + . . . + L (4)
where the last term L is A 1 cos (φ -A 0 ) if n is odd, and A 0 if n is even.

Mass and center of mass

It is well-known and easily demonstrated that for each projection φ, Π 0 (φ) is the mass of µ:

Π 0 (φ) = R p φ (s) ds = R 2 µ(s θ φ + l ζ φ ) dl ds = R 2 µ( x) d x (5)
Only slightly less well-known is the fact that the center of mass c φ of the φprojection p φ is equal to the φ-projection, c • θ φ , of the center of mass c ∈ R 2 of µ. Indeed

c φ = R sp φ (s) ds R p φ (s) ds = Π 1 (φ) Π 0 (φ) = R s R µ(s θ φ + l ζ φ ) dl ds R R µ(s θ φ + l ζ φ ) dl ds = R 2 ( x • θ φ )µ( x) d x R 2 µ( x) d x = R 2 xµ( x) d x • θ φ R 2 µ( x) d x = c • θ φ 2.

Projection index and ray index

Suppose p = Rµ and now let p Σ (φ, s) . . = p(-φ, s). The corresponding moments are Π Σ n (φ) = Π n (-φ) which are also of the required form (Eq. ( 3)), so they correspond to some function µ Σ (i.e. p Σ = Rµ Σ ). We easily verify that µ Σ (x 1 , x 2 ) = µ(x 1 , -x 2 ), which says that if the projection angle is reversed, the reconstructed function is reflected about the x 1 -axis. Now consider p (δ,0) (φ, s) . . = p (φ + δ, s), and we observe that the corresponding moments Π (δ,0) n (φ) = Π n (φ + δ) are also of the required form (Eq. ( 4)). The corresponding function µ δ is obviously just the rotated version

µ δ (x 1 , x 2 ) = µ(x 1 , x 2 ) with x 1 = x 1 cos δ -x 2 sin δ, x 2 = x 1 sin δ + x 2 cos δ.
This time, shifting the ray index by a constant (instead of the projection index), we define p (0,δ) (φ, s) . . = p (φ, s + δ). We find Π (0,δ) n (φ) = Π n (φ) + nδΠ n-1 (φ)+n(n-1)/2δ 2 Π n-2 (φ)+. . .+δ n Π 0 (φ) which is a polynomial in sin φ and cos φ but it is not homogeneous. So the projections p (0,δ) are not consistent and do not correspond to any function, which agrees with our intuition that the projections are no longer correctly aligned with each other. However, a sinusoïdal shift of the ray index is consistent:

∀ v ∈ R 2 , p (φ, s + v 1 cos φ + v 2 sin φ) is as consistent as p (φ, s). Indeed let µ v . . = µ( x + v) then Rµ v (φ, s) = Rµ(φ, s + v • θ φ ).

Parallel Linograms

We examine the parallel linogram case as a stepping stone to fan-beam linograms. The parallel linogram l(u, q) is defined by

l u (λ) . . = l(u, q) . . = Lµ(u, q) . . = R µ (q, 0) + l(-u, 1) dl. ( 6
)
The parallel linogram projections are of the form l u where the linear projection index u specifies the orientation of the projection. See Fig. 2. Consistency conditions for parallel linograms are easily derived from the Helgason-Ludwig conditions, since the lino-grams just describe the same projections using different variables [START_REF] Clackdoyle | Data consistency for linograms and planograms[END_REF]. Let J n (u) be the n th moment of l u :

J n (u) . . = R l u (q)q n dq (7)
then l = Lµ for some object µ if and only if, ∀n, J n (u) is a polynomial of degree at most n, i.e. J n (u) is of the form

J n (u) = b 0 + b 1 u + . . . + b n u n (8)

Mass and center of mass

It can be shown directly that for each projection u , J 0 (u) is the mass of µ. This property is not obvious because l(u, q) = cos φ p(φ, s) (where u = tan φ and s = q cos φ), and the scaling factor cos φ(= 1/ √ 1 + u 2 ) would suggest the mass is related to J 0 via some function of u . However, at larger u , the sampling in q is denser, and the combined effect cancels to yield the simple relation

J 0 (u) = R l u (q) dq = R R µ (q -lu, l) dl dq = R 2 µ ( x) d x (9)
where we have made the change of variables x 1 = q -lu and x 2 = l.

The center of mass of the u-projection l u is moreover

c u . . = R l u (q)q dq R l u (q) dq = J 1 (u) J 0 (u) (10)
J 1 (u) = R l u (q)q dq = R R qµ (q -lu, l) dl dq = R 2 (x 1 + x 2 u)µ ( x) d x (11)
with the same change of variable as in [START_REF] Clackdoyle | Data consistency for linograms and planograms[END_REF] and so q = x 1 + ux 2 . From (10), ( 9) and (11) we have

c u = c 1 + uc 2 . (12)
Now, for an arbitrary point (x 1 , x 2 ), the "u-projection" (the corresponding value of q) is x 1 + ux 2 = (x 1 , x 2 ) • (1, u), see Fig. 2. Thus we clearly see that the center of mass of the u-projection l u is the u-projection of the center of mass (of µ).

In terms of the moments:

J 1 (u) J 0 (u) = c • (1, u). (13)

Projection index and ray index

In this subsection, we assume l = Lµ. Letting l Σ (u, q) . . = l(-u, q), we easily see that the corresponding moment J Σ n (u) = J n (-u) is also a polynomial of the same degree as J n (u). We also easily see that l Σ = Lµ Σ , exactly as in the conventional Radon transform: reversing the sign of the projection index will reflect the object function about the x 1 -axis. Now let l δ (u, q) . . = l(u+δ, q) ; the corresponding moments J δ n (u) = J n (u+δ) are obviously also polynomials of degree n or less, so there again exists an object µ δ (not the same as the previous µ δ ) such that l δ = Lµ δ . We have established that

µ δ ( x) = µ (A δ x) where A δ . . = 1 δ 0 1 (14)
which is a linear shear mapping of µ with shear factor -δ. Indeed

Lµ δ (u, λ) = +∞ 0 µ δ (λ -lu, l) dl = +∞ 0 µ (λ -lu + lδ, l) dl thus Lµ δ (u, λ) = Lµ(u -δ, λ) (15)
For the case of the translated ray index, we define l(u, q) . . = l(u, q + δ). The corresponding moments satisfy J n (u) = J n (u) + nδJ n-1 (u) + . . . + δ n J 0 (u), which is indeed a polynomial in u of degree n or less. So there exists some µ such that l = L µ. We find µ(x 1 , x 2 ) = µ(x 1 + δ, x 2 ), a simple translation of µ by δ(1, 0). But this is a particular case of the following: with

µ v (x) . . = µ(x + v) be a translation of µ of v = (v 1 , v 2 ) ∈ R 2 then we have Lµ v (u, q) = R µ (q + v 1 , v 2 ) + l(-u, 1) dl = R µ (q + v 1 , v 2 ) + (l + v 2 )(-u, 1) dl = Lµ(u, q + v 1 + v 2 u) (16)

Fan-Beam Linograms

For fan-beam linograms, we define the linogram data d(λ, t) by d(λ, t) . . = (Dµ)(λ, t)

. . = +∞ 0 µ (λ, D) + l(t -λ, -D) dl ( 17
)
where the projection parameter λ indicates the location (λ, D) of the fan-vertex along the line x 2 = D, and the ray-variable t indicates the absolute location along the x 1 -axis, which can be considered as a fixed detector. See Fig. 3. Note that this definition is fundamentally different from that in [START_REF] Clackdoyle | Data consistency for linograms and planograms[END_REF] where the ray variable indicates a point relative to the fan-vertex, as if the detector and fanvertex (e.g. x-ray source) were a single unit. Consistency conditions for d(λ, t) can be found in [START_REF] Clackdoyle | Necessary and sufficient consistency conditions for fanbeam projections along a line[END_REF]. Let K n (λ) be the n th moment of d(λ, t):

∀n ∈ N, K n (λ) = R d(λ, t)t n dt (18)
then d = Dµ for some µ if and only if, ∀n ∈ N, K n (λ) is a polynomial of degree at most n. In [START_REF] Clackdoyle | Necessary and sufficient consistency conditions for fanbeam projections along a line[END_REF], it is shown that

K n (λ) = n k=0 α n-k,k λ k (19)
where

α n-k,k = n k R 2 µ(x 1 , x 2 ) x n-k 1 D n-k (-x 2 ) k (D -x 2 ) n+1 dx 1 dx 2 . ( 20
)
Because the compact support of µ is included in R × (-∞, D), the singularity term (D -x 2 ) -(n+1) does not affect the integral.

Mass and center of mass

From ( 19) and (20) we derive that

K 0 (λ) = R 2 µ( x) D -x 2 d x = α 0,0
i.e. for each λ-projection, the constant K 0 (independent of λ) equals the total mass of the weighted object function µ( x) D-x2 . Note that the object lies under the line x 2 = D and the weight is the inverse distance from x = (x 1 , x 2 ) to the line x 2 = D.

We also have

K 1 (λ) = α 1,0 + α 0,1 λ (21)
where

α 1,0 = D R 2 x 1 µ( x) (D -x 2 ) 2 d x (22) α 0,1 = - R 2 x 2 µ( x) (D -x 2 ) 2 d x (23)
Let us define µ W , the function µ weighted by

1 (D-x2) 2 µ W ( x) . . = µ( x) (D -x 2 ) 2
We note that

K 0 = R 2 µ W ( x) d x (D -c W 2 )
where c W = (c W 1 , c W 2 ) the center of mass of µ W , i.e.

c W 1 . . = R 2 x 1 µ W ( x) d x R 2 µ W ( x) d x and c W 2 . . = R 2 x 2 µ W ( x) d x R 2 µ W ( x) d x It can easily be shown that R 2 µ W ( x) d x = α 0,0 -α 0,1 D c W 1 = α 1,0 α 0,0 -α 0,1 c W 2 = - Dα 0,1 α 0,0 -α 0,1
From Eq. ( 21), ( 22) and ( 23)

K 1 (λ) = (D, -λ) • c W R 2 µ W ( x) d x. (24)
Now, for an arbitrary point (x 1 , x 2 ), the λ-projection of this point is given by t

λ (x 1 , x 2 ) = (x 1 , x 2 ) • (D, -λ)/(D -x 2 ). Indeed, t λ (x1,x2)-λ x1-λ = D D-x2
, see Fig. 4. Therefore, the center of mass of the λ-projection is the λ-projection of the center of mass of the weighted object µ W . Expressed with moments:

K 1 (λ) K 0 = c W • (D, -λ)/(D -c W 2 ). ( 25
) D x 2 (λ, D) x 1 x 1 x 2 D x = (x 1 , x 2 ) D -x 2 λ x 1 -λ t λ (x 1 , x 2 ) t λ (x 1 , x 2 ) -λ Figure 4: t λ (x 1 , x 2 ) is λ-projection of the point (x 1 , x 2 ). Thus t λ (x1,x2)-λ x1-λ = D D-x2 .

Projection index and ray index

Following the pattern of the previous sections, we define d

Σ (λ, t) = d(-λ, t); d δ (λ, t) = d(λ + δ, t); d(λ, t) = d(λ, t + δ). Assuming d(-λ, t) = (Dµ)(λ, t),
then all three modified fan-beam linograms are also consistent, by the same Similarily, the center of mass of the fan beam linogram projection d λ is the λprojection of the center of mass of µ W where µ W ( x) = µ( x) (D-x2) 2 . Offsetting the projection index for the Radon transform corresponds to rotating the object, whereas for linograms, it corresponds to performing a shear transformation on the object, see Fig. 5.

We have recalled for the 2D Radon data p and shown for l (parallel linogram data) and d (fan-beam linogram data) that • ∀ v ∈ R 2 , p φ, s + v • θ φ , p (-φ, s) and p (φ + δ, s) are as consistent as p (φ, s)

• ∀ v ∈ R 2 , l u, λ + v • (1, u) , l (-u, λ), l (u + δ, λ) are as consistent as l (u, λ).

• d(λ, t + δ), d(-λ, t) and d(λ + δ, t) are as consistent as d(λ, t).

We have given the respective transforms on µ to obtain the respective consistent data, assuming respectively p = Rµ, l = Lµ, d = Dµ. As in [START_REF] Basu | Uniqueness of tomography with unknown view angles[END_REF], this information (δ, v, the "direction" of the projection acquisition, i.e., the global sign of the respective projection parameter φ, u, λ) can not be estimated from the DCCs.

1 OFigure 1 :

 11 Figure 1: 2D parallel geometry. The line of integration is the dashed blue line s θ φ + R ζ φ .

1 c 1 + c 2 u = c u c u φ c 2 u 1 Figure 2 :

 1112 Figure 2: Linogram geometry. The line of integration is the dashed blue line (λ, 0) + R(-sin φ, cos φ) or equivalently (λ, 0) + R(-u, 1) with u = tan φ (up to the scaling factor cos φ). λ c = c 1 + c 2 u is the linogram projection of the point c.

Figure 3 :

 3 Figure 3: 2D fan-beam linogram tomography. The (half) line of integration (λ, D) + R + (x -λ, -D) is the dashed blue (half) line. The source position is the red point at (λ, D). The (virtual) detector position is the blue square at (t, 0).

Figure 5 :

 5 Figure 5: Example of effect of constant offset applied to the projection variable. Left: µ(x 1 , x 2 ) is the standard Shepp-Logan phantom. Middle: Rotation due to Radon projections offset by a constant shift. Right: Shear transformation due to linogram projections offset by a constant shift.

This work was supported in part by the "Fonds unique interministériel" (FUI) and the European Union FEDER in Auvergne Rhône Alpes (3D4Carm project).

arguments as in section 3.2. There exists therefore, µ Σ , µ δ , µ such that d Σ = Dµ Σ , d δ = Dµ δ , d = D µ. We find

, so a constant shift of either the projection index or the ray variable corresponds to a shear of the object.

The expression for µ Σ is more complex. We show

Discussion

We have studied five well-known properties of the Radon transform in the context of parallel and fan-beam linograms, and seen remarkable differences and similarities. For example, the center of mass of the 2D Radon parallel projection p φ , respectively the parallel linogram projection l u , is the φ-projection, respectively the u-projection, of the center of mass of the measured object µ.


Mauro D ' Onofrio

Daniela Bettoni

Anna Lia Longinotti

Anelise Audibert
email: anelise.audibert@obspm.fr

Françoise Combes

Santiago García-Burillo

Philippe Salomé

Probing the Gas Fueling and Outflows in Nearby AGN with ALMA

Keywords: active galactic nuclei, extragalactic astronomy, evolution of galaxies, galaxies, spectroscopy

Feeding and feedback in AGN play a very important role to gain a proper understanding of galaxy formation and evolution. The interaction between activity mechanisms in the nucleus and its influence in the host galaxy are related to the physical processes involved in feedback and the gas fueling of the black hole. The discovery of many massive molecular outflows in the last few years have been promoting the idea that winds may be major actors in sweeping the gas out of galaxies. Also, the widely observed winds from the central regions of AGN are promising candidates to explain the scaling relations (e.g., the black hole-bulge mass relation, BH accretion rate tracking the star formation history) under the AGN feedback scenario. Out goal is to probe these phenomena through the kinematic and morphology of the gas inside the central kpc in nearby AGN. This has recently been possible due to the unprecedented ALMA spatial resolution and sensitivity. We present results on NGC7213 and NGC1808, the latter is part of a new ALMA follow-up of the NuGa project, a previous high-resolution (0.5-1 ′′ ) CO survey of low luminosity AGN performed with the IRAM PdBI.

INTRODUCTION

The fueling of SMBH at the center of galaxies and the subsequent feedback from its active nuclei, are among the key processes to understand the concerted growth of galaxies and BH. AGN feedback is invoked to explain the formation of massive galaxies when we compare the luminosity function of galaxies to theoretical simulations based on the cold dark matter models [START_REF] Silk | The current status of galaxy formation[END_REF]. Winds and outflows produced by the AGN can eject or heat the gas, terminate the star formation and through the lack of fuel for accretion, quench the black hole activity. Recent discovery of many massive molecular outflows (e.g., [START_REF] Feruglio | Quasar feedback revealed by giant molecular outflows[END_REF][START_REF] Fischer | Herschel-PACS spectroscopic diagnostics of local ULIRGs: conditions and kinematics in Markarian 231[END_REF][START_REF] Alatalo | Discovery of an active galactic nucleus driven molecular outflow in the local early-type Galaxy NGC 1266[END_REF][START_REF] Sturm | Massive molecular outflows and negative feedback in ULIRGs observed by herschel-PACS[END_REF][START_REF] Dasyra | Cold and warm molecular gas in the outflow of 4C 12.50[END_REF][START_REF] Veilleux | Fast molecular outflows in luminous galaxy mergers: evidence for quasar feedback from herschel[END_REF][START_REF] Cicone | Massive molecular outflows and evidence for AGN feedback from CO observations[END_REF][START_REF] Dasyra | Heating of the molecular gas in the massive outflow of the local ultraluminous-infrared and radio-loud galaxy 4C12.50[END_REF][START_REF] García-Burillo | Molecular line emission in NGC 1068 imaged with ALMA. I. An AGN-driven outflow in the dense molecular gas[END_REF] have been promoting the idea that winds may be major actors in sweeping the gas out of galaxies, in agreement with theoretical predictions of AGN driven winds models [START_REF] Faucher-Giguère | The physics of galactic winds driven by active galactic nuclei[END_REF]Zubovas andKing, 2012, 2014). However, it is still difficult to distinguish the origin of the outflows, whether they are AGN-driven or starburst-driven.

On the other hand, AGN are fueled by accretion of material onto the BH and the gas component can form stars on its way to the BH (García-Burillo and Combes, 2012;[START_REF] Combes | ALMA reveals the feeding of the Seyfert 1 nucleus in NGC 1566[END_REF]. It is important to study the efficiency of angular momentum transport in galaxy disks in order to understand how the star formation and nuclear activity are fueled and what are the timescales involved, since both feeding process rely on a common cold gas supply, but in very different periods of time (∼10 5 year for BH growth and ∼10 7-9 year for star formation; [START_REF] García-Burillo | Gas flows in galactic nuclei: observational constraints on BH-galaxy coevolution[END_REF]. In addition to this scenario, one of the outstanding problems is to identify the mechanism that drives gas from the disk toward the nucleus, removing its large angular momentum and forming large non-axisymmetric perturbations, such as bars or spirals.

This work aims to better understand the galaxies chemical and thermal evolution by studying in detail the physical processes acting on the nucleus of a sample of nearby AGN. Probing AGN feeding and feedback phenomena through the kinematic and morphology of the gas inside the central kpc have recently been possible due to the unprecedented Atacama Millimeter/Submillimeter Array (ALMA) spatial resolution ( 0.5 ′′ ) and sensitivity. Since there are only a few AGN observed with high spatial resolution, we investigate the nuclear molecular emission in a sample of nearby AGN making use of spectroscopic techniques.

METHODOLOGY

By applying the velocity channel maps on the dense molecular gas tracers, e.g., HCN, HCO + , CS, and mid transition of CO, we can access the gas kinematics. To detect possible inflows or outflows, we can compare the CO rotation curve with a rotational velocity model based on the Hα kinematics and then subtract the model to analyse the velocity residuals in order to search for blue and redshifted patterns. Another possibility is to explore the current models for the rotation curve, when previous Hα or H I are not available in the literature. One of this model is described by [START_REF] Bertola | Testing the gravitational field in elliptical galaxies -NGC 5077[END_REF], assuming the gas is on circular orbits in a plane, υ c = Ar/(r 2 + c 2 ) p/2 , where for p = 1 the velocity curve is asymptotically flat and p = 3/2 the system has a finite total mass, therefore we expect 1 ≤ p ≤ 3/2. The observed radial velocity at a position (R, ) on the plane of the sky can be described as:

υ mod (R, ) = υ sys + ARcos( -0 )sin(θ )cos p (θ ) R 2 sin 2 ( -0 ) + cos 2 (θ )cos 2 ( -0 ) + c 2 cos 2 (θ ) p/2 (1)
where θ is the inclination of the disk (with θ = 0 for a faceon disk), 0 is the position angle of the line of nodes, v sys is the systemic velocity and R is the radius and A, c, and p are parameters of the model. In the cases of NGC 7213 and NGC 1808, we have used the [START_REF] Bertola | Testing the gravitational field in elliptical galaxies -NGC 5077[END_REF] model to search for non-circular motions in their observed velocity field.

RESULTS

NGC 7213

We report ALMA Cycle 1 observations of CO(2-1) at 229.2 GHz of NGC 7213, an early-type SA(s)a galaxy harboring a Seyfert 1/LINER. Previous neutral and ionized gas observations suggest the presence of a warped gaseous disk that could be the result of a merger. It presents a giant Hα filament and two HI tidal tails, confirming to be a highly disturbed interaction system [START_REF] Hameed | The discovery of a giant Hα filament in NGC 7213[END_REF]. Stellar kinematics is dominated by non-circular motions, along two spiral arms [START_REF] Schnorr-Müller | Gas inflows towards the nucleus of the active galaxy NGC 7213[END_REF]). In the central ∼3 kpc, the CO map reveals a a widely distributed multiple-arm spiral structure, tracing the dusty spiral arms, as can be seen in the HST image (Figure 1). There is an offset peak located at 200 pc north from the center, meaning that the peak of the CO emission is not concentrated in the very center.

In general, the velocity field in the left panel of Figure 2 is welldescribed by rotation, presenting some slightly perturbations along the minor axis (at 10 ′′ ) probably due to streaming motions from the spiral arms. By applying the [START_REF] Bertola | Testing the gravitational field in elliptical galaxies -NGC 5077[END_REF] model to generate a 2-D velocity model that minimizes the residuals, we find that the position angle is -18 • and the systemic velocity should be corrected in 30 km/s. The residuals do not indicate hints of a significant blue and/or red-shifted components and this result indicates that probably there is no molecular outflow in NGC 7213.

NGC 1808

The "hot spot" galaxy NGC 1808 is remarkable for its very dusty appearance and especially for its well-known system of dark radial filaments. NGC 1808 is a Starburst/Seyfert 2 galaxy located at 9.1 Mpc and it is classified as a SAB(s)a. Accordingly to [START_REF] Reif | An HI survey of southern galaxies[END_REF], it has an inclination of 57 • . Atomic gas traced with H I is concentrated in the galactic bar, disk, and a warped outer ring, indicating a tidal interaction in the past with the neighbor galaxy NGC 1792 [START_REF] Koribalski | Gas streaming along the bar in NGC 1808. A combination of high-resolution Hα and HI data[END_REF]. It presents prominent polar dust lanes and gas outflow emerging from the nucleus, as revealed in optical studies that indicate an outflow of dust and neutral and ionized gas (as seen in Hα, [N II], Na I D, and H I by [START_REF] Koribalski | A comprehensive study of the peculiar spiral galaxy NGC 1808. II -VLA H I line observations[END_REF][START_REF] Phillips | Nuclear and large-scale outflow in NGC 1808[END_REF]. In particular, Na I D is seen blueshifted in absorption and redshifted in emission and B-R images show that the dust plumes associated with the outflow reach ∼3 kpc above the plane. Near-infrared integralfield spectroscopy with SINFONI by [START_REF] Busch | Star formation and gas flows in the centre of the NUGA galaxy NGC 1808 observed with SINFONI[END_REF] showed a large gas reservoir and a disturbed gas velocity field that shows signs of inflowing streaming motion in the central ∼100 pc.

We report ALMA Cycle 3 observations of CO(3-2), CS(7-6), HCN(4-3), and HCO + (4-3) in band 7 with a resolution of 0.27 ′′ . Figure 3 shows that the CO emission follows the star-forming central 500 pc ring and the very center has a spiral trailing structure, that is also observed in CS, HCN, and HCO + .

From the velocity map in Figure 4, the dominant velocity feature appears to be due to circular rotation in the disk. After subtracting a model for the velocity field by [START_REF] Bertola | Testing the gravitational field in elliptical galaxies -NGC 5077[END_REF], we do not find significant patterns in the residuals and if there is a presence of an molecular outflow in the galaxy, it will be difficult to disentangle from the dominant circular motion in the 1st moment distribution. We need to explore in details the position-velocity diagrams along the dust lanes observed in the HST images. Additionally, [START_REF] Salak | Gas dynamics and outflow in the barred starburst galaxy NGC 1808 revealed with ALMA[END_REF] presented evidence of a molecular gas outflow seen in CO(1-0) from the nuclear starburst region (r 250 pc) that could be only detected in the position-velocity diagram. The outflow has a maximum velocity of υ out ∼180 km/s and mass outflow rate between 1 and 10 M ⊙ /year, comparable to the total star formation rate in the starburst 500 pc region, that corresponds to SFR ∼ 5M ⊙ /year in the "hot spots." The kinematics of our CO(3-2) observations do not show strong outflows at this scale. Certainly, the outflow is more significant at larger scales. A further analysis of the molecular gas and the outflow in NGC 1808 will be present in a forthcoming paper (Audibert et al., in preparation).

PERSPECTIVES

We are going to apply the same methodology to search for outflows and inflows in a modest sample of nearby Seyfert/LINERs galaxies that are part of a new ALMA followup of the-NUclei of GAlaxies (NuGa) project, a previous CO survey of low luminosity AGN performed with the IRAM PdBI. The new ALMA observations have the remarkable spatial resolution of ∼0.06-0.09 ′′ .

FIGURE 1 |

 1 FIGURE 1 | In the left and middle panels we show an unsharp-masked F606W HST image and the original image with CO(2-1) contours overlaid, respectively. Right: CO(2-1) intensity map of NGC 7213 clipped at >5σ . The green circle represents a 30 ′′ diameter.

FIGURE 2 |

 2 FIGURE 2 | Left: Velocity map of NGC 7213 clipped at >5σ . In the middle panel the best fit model using the Bertola et al. (1991) function and the residuals, after subtracting the model from the data, are shown on the right.

FIGURE 4 |

 4 FIGURE 4 | Left: Velocity map of NGC 1808 clipped at >5σ . In the middle panel the best fit model using the Bertola et al. (1991) function and the residuals, after subtracting the model from the data, are shown on the right.

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 2 December 2017 | Volume 4 | Article 58

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 3 December 2017 | Volume 4 | Article 58

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 4 December 2017 | Volume 4 | Article 58

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 5 December 2017 | Volume 4 | Article 58

ACKNOWLEDGMENTS AA thanks the organizers of the "Quasars at All Cosmic Epochs" conference for support and the referees for useful comments. AA would like to acknowledge the financial support from the Brazilian scholarship program Science Without Borders-CNPq (scholarship reference 234043/2014-8). This paper makes use of the following ALMA data: ADS/JAO.ALMA# 2012.1.00474.S (PI: N. Nagar) and ADS/JAO.ALMA# 2015.1.00404.S (PI: FC). ALMA is a partnership of ESO (representing its member states), NSF (USA), and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ. We used observations made with the NASA/ESA Hubble Space Telescope, and obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESA), and the Canadian Astronomy Data Center (CADC/NRC/CSA). This research has made use of the NASA/IPAC Extragalactic Database (NED) and of the HyperLeda database.

AUTHOR CONTRIBUTIONS

AA has written the article; AA and FC have contributed to reduce the data; FC and SG-B have written the proposal; and all four authors have contributed to interpret the data and comment the paper.

Conflict of Interest Statement:

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2017 Audibert, Combes, García-Burillo and Salomé. This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.