A Oukili

J.-C Nunes

Y Chen

L Luo

C Toumoulin

OBJECT-BASED 3D BINARY RECONSTRUCTION FROM SPARSE PROJECTIONS IN CONE BEAM CT: COMPARISON OF THREE PROJECTION OPERATORS

Keywords: cone-beam geometry, projection operator, 3D iterative Reconstruction, object-based reconstruction, level set

We present herein a level set approach to the X-ray tomography problem with sparse projection data and study the impact of the projection operator on the binary reconstruction accuracy and computation time. The comparison is carried out on three projectors: the Separable Footprint (Trapeze-Trapeze, SF-TT) [3], a classical Raydriven (RD) and a Simplified version of the Distance-Driven (SDD) projector respectively. The performance, are evaluated for each operator, on a binary 3D Shepp-Logan phantom by varying the number of projections from 5 to 13, and considering noise free and noisy cone beam projection data.

INTRODUCTION

Cone Beam Computed tomography is becoming more and more attractive due its capability to integrate the intervention room and provide intraoperative imaging. The imaging system is composed of a gantry that rotates around the patient and on which an X-ray source and flat detector are fixed. During rotation, a sequence of image projections can be obtained (between 100 and 200), which are then exploited to get a 3-D reconstruction of the organs of interest. Many studies have been conducted to reduce radiation dose and the acquisition number [START_REF] Hu | L0 constrained sparse reconstruction for multi-slice helical CT reconstruction[END_REF]. We suggest here reducing the patient dose by limiting the projection number during a scan. As a consequence, the 3-D reconstruction from a few number of projections becomes strongly ill-posed. Works reported in the literature makes use of iterative methods to solve this problem [START_REF] Hu | L0 constrained sparse reconstruction for multi-slice helical CT reconstruction[END_REF][START_REF] Kim | Forward-projection architecture for fast iterative image reconstruction in X-ray CT[END_REF].The latter relies on the computation of forward and backward projections and the quality of this reconstruction is highly dependent on the accuracy of the corresponding system matrix. Basic operators include voxel [START_REF] Joseph | An Improved Algorithm for Reprojecting Rays Through Pixel Images[END_REF][START_REF] Zhuang | Numerical Evaluation of Methods for Computing Tomographic Projections[END_REF], splatting [START_REF] Long | 3D forward and backprojection for X-ray CT using separable footprints[END_REF], ray-driven (RD) [START_REF] Siddon | Fast calculation of the exact radiological path for a three-dimensional CT array[END_REF], Separable Footprint (Trapeze-Trapeze, SF-TT) [START_REF] Long | 3D forward and backprojection for X-ray CT using separable footprints[END_REF] and distance driven (DD) [START_REF] Zeng | A ray-driven backprojector for backprojection filtering and filtered backprojection algorithms[END_REF][START_REF] De Man | Distance-driven projection and backprojection in 3D[END_REF] approaches. Our contributions concern two phases (presented in Section 2): Simplified version of the Distance-Driven (SDD) and a level set approach to perform the 3D binary tomographic reconstruction. Indeed, in binary case, object-based reconstruction approaches using the level set method, have shown promising results compared to classical methods since this simultaneous segmentation and reconstruction leads to improved results [START_REF] Gaullier | Introducing shape priors in object-based tomographic reconstruction[END_REF]. We studied in section 3 the binary reconstruction obtained from each projection operator (SF-TT, RD and SDD), in varying the number of projections from 5 to 13, and considering noise free and noisy cone beam projection data. 3D object f is located between the X-ray source S and the flatdetector which is at a distance DSD from S. The source-detector rotates around Z-axis on which the 3D object is fixed (Fig. 1). In order to perform the geometric calculation, we define a system of three-dimensional Cartesian coordinates, (o, i, j, k), such as i = ∆x, j = ∆y, and k = ∆z, and another planar coordinate system fixed on the detection grid, consisting of two orthogonal axes, S and T , such as T is parallel to OZ. (∆x, ∆y, ∆z) and (∆s, ∆t) represent respectively the voxel and pixel sizes.

After the referential definition of rotational acquisition system, first begin by determining the exact value of a projection point P (s, t; θ) (eq.2), taking into account the cone-beam geometry, and the source position P0:

P0 =   -DSO sin (θ) DSO cos (θ) 0   (1)
where, DSO denotes the source to rotation center, and θ denotes angle of the source point counter-clockwise from the Y axis.

P (s, t; θ) = χ(s,t,θ) f (x, y, z)dχ (2)
for which, the integral is along the line segment χ ( eq.3 ), and (s, t) denote the detector coordinates.

χ(s, t, θ) = { P0 + l e : l ∈ [0, Lp]} Lp = D 2 SD + s 2 + t 2 (3)
where e is the direction of the line χ(s, t, θ) connecting the source S to the point P :

e = P -P 0 P -P 0 =   sin ϕ cos β -cos ϕ cos β sin β   ( 4
)
where

β = -arctg( t √ s 2 +D 2

SD

) and ϕ = arctg( s D SD ) + θ denotes respectively the azimuthal and polar angle of the ray from P0 to P . After determining the mathematical expression of a detection point, we identify the value detected by each pixellic surface ḡθ [s k , t l ] as:

ḡθ [s k , t l ] = ∆s ∆ t h(s k -s, t l -t)P (s, t; θ)dsdt (5)
where, (s k ,t l ) represents the centers of detector cell for: k = 0, ..., Ns-1 , l = 0, ..., Nt-1 , s (k+1) -s k = ∆s, and t (l+1)s l = ∆t. h is the detector blur which depends only on coordinates (s, t). Taking into consideration the continuous representation of our 3D discrete object (eq. 6) (which will be defined as a weighted sum of basis function b ∆ ), we can calculate cone-beam footprint q(s, t; θ; n) of the n th basis function:

f ( x) = n f [ n]b ∆ ( x -c [ n]) (6)
where c [ n] denotes the center of the n th basis function.

q(s, t; θ; n) = χ(s,t;θ) b ∆ ( x -c [ n]) dχ (7)
Finally, for a projection angle θ, the coefficient linking the n th basis function with a pixel indexed by (s k , t l ) is determined as:

a θ [s k , t l ; n] = ∆s ∆ t h(s k -s, t l -t)q(s, t; θ; n)dsdt (8)
However, the exact calculation of a θ [s k , t l ; n] remains difficult and computationally expensive, hence the necessity to make approximations. The two following sections explain the two different approximation methods.

Separable Footprint Projector : Trapeze-Trapeze (SF-TT)

For the SF-TT method proposed by Long et al. [START_REF] Long | 3D forward and backprojection for X-ray CT using separable footprints[END_REF], we consider a simple model for the basis function b ∆ :

b ∆ ( x -c [ n]) = b0(( x -c [ n])∅ ∆) b0( x) = Π(x) × Π(y) × Π(z) (9) Z Y X s T Voxel n s0 tl X Y Z X S P(s k ,t l ) D V1 V 2 V 1 V 2 e Detector s1 s2 s3 S T t0 t1 t2 t3 sk A s A t = D A s A t
h(s, t) = 1 ∆s∆t Π s ∆s Π t ∆t (10)
Inspired by the shape of the true footprint q(s, t; θ; n) of the basis functions (eq.9) [START_REF] Long | 3D forward and backprojection for X-ray CT using separable footprints[END_REF], with exploitation of the overlap area with the surface of each pixel, Yong Long et al. [START_REF] Long | 3D forward and backprojection for X-ray CT using separable footprints[END_REF] have established a robust approximation of q (Fig. 2) :

q(s, t; θ; n) ≈ D × trap(s; s0, s1, s2, s3) × trap(t; t0, t1, t2, t3) (11) where trap(a; b0, b1, b2, b3) is trapezoid function with b0, b1, b3 and b4 represent the vertices of the trapeze. D is the voxel-ray-dependent amplitude [START_REF] Long | 3D forward and backprojection for X-ray CT using separable footprints[END_REF] :

D = ∆x |cos(β)| × max(|cos(ϕ)| , |sin(ϕ)|) (12)
The combination of the equations eq.8, eq.10, and eq.11, allows the computation of a θ [s k , t l ; n] :

a θ [s k , t l ; n] = ∆s ∆ t h(s k -s, t l -t)q(s, t; θ; n)dsdt = D × T (s k -∆s 2 , s k + ∆s 2 ) × T (t l -∆ t 2 , t l + ∆ t 2 )
(13) where T represents the overlap area with the pixel (the surface AS and At in Fig. 2).

Simplified Distance Driven Projector

The coefficient a θ [s k , t l ; n] can be defined as the likelihood that one photon emitted by the voxel n will be detected by the pixel (s k , t l ). Inspired by this definition, and the Distance Driven method [START_REF] De Man | Distance-driven projection and backprojection in 3D[END_REF], we proceed in three steps : 1) For each voxel n, we compute the conical projections of the eight cubic vertices (green dots in Fig. 3).

2) We draw the smallest rectangle enclosing these projection points, and we can assume that the probability that a photon emitted by this voxel n will be detected in the rectangular surface (AT in Fig. 3) which is equal 1.

3) We calculate the overlap surface between each pixel and the rectangular area (AP in Fig. 3), and the coefficients of the projection matrix is inferred as a area ratio as : a θ [s k , t l ; n] = AP / AT

Model-Based Binary Reconstruction

To evaluate these computation methods of projection matrices and, in particular, their impact on reconstruction, we chose methods based only on minimizing a cost function which is a data attachment term without regularization:

E d = 1 2 p -Af 2 .
Object-oriented methods assume that the reconstruction consists of a background region Ωout and a foreground region, Ωin, representing the object [START_REF] Gaullier | Introducing shape priors in object-based tomographic reconstruction[END_REF]. Consequently, in the case of a binary reconstruction, the data attachment term can be rewritten in as :

E d = 1 2 p -finAδΩ in -foutAδΩout (14)
where fint and fout represent the intensity value of the background and the object respectively, and δΩ i is the characteristic function of Ωi.

To optimize E d , we adopt deformable models (precisely the level set methods), for which the velocity field V d can be calculated by applying derivation tools of the form :

V d = (fout -fin)A T (p -Af ) (15)
The use of level set function make easy the determination of different regions Ωi and their characteristic functions δΩ i . The evolution equation is defined as : ∂ψ ∂t = V d |∇ψ| fout and fin depend of δΩ i and they are updated after each evolution of level-set function ψ [START_REF] Gaullier | Introducing shape priors in object-based tomographic reconstruction[END_REF].

RESULTS AND DISCUSSIONS

Projection Results

We simulated the X-ray cone-beam projections of a binary 3D Shepp-Logan phantom, which includes a sphere, a rectangular parallelepiped, and two ellipsoids (Fig. In accordance with these acquisition parameters, projections are generated by implementing SF-TT, SDD and RD projectors, for different projection angles θ, such as θ ∈ -π 2 , + π 2 . In order to study the noise influence on the reconstruction results, and have more realistic projection images, an additive Gaussian noise is introduced with different values of SN RV AR ( Fig. 4.II).

Binary Reconstruction Results

The binary reconstruction is based on Object-oriented method, for which we have chosen deformable models to minimize a cost function based on data attachment term (eq.14). To quantify the quality of these reconstructions, we chose the criterion NMSE (Normalized Mean Square Error). Firstly, we studied the convergence speed of each reconstruction, while varying projection operators (SF-TT, SDD and RD). For this first test series, we plot the NMSE of reconstruction for the first 100 iterations, when using 13, 7 or 5 projections (respectively columns A, B and C in Fig. 5), and different noise levels SN RV AR, without noise, 25dB, and 20dB (Lines 1, 2 and 3 in Fig. 5). For each reconstruction, we chose the same initial coarse object ( binary hollow sphere), and we initialized fout and fin at 1 and 3. In a second step, we evaluated the reconstruction accuracy by varying the noise level from SN RV AR =10 to 100 db, and the projection number (Fig. 6.II). For this test series, we chose an initial object (Fig. 6.I.B) which presents a rough approximation of our object (binarization of a Least Mean Square Error Reconstruction) after a few iterations), to be sure to achieve the optimal reconstruction.

Discussion

By analyzing Fig. 5, we can say that in the case of level set initialization with a coarse object, and in the absence of noise, our binary reconstruction converges faster when using SDD projector, compared to SF-TT and RD projectors, and this speed appears faster when we decrease the number of projections. The fact of adding noise reduces this speed difference. Should note here that one iteration is defined as a reconstruction after having used whole projections (13, 7 or 5), and the jump that appears in NMSE curves (line 1) is due to the reconstruction of small structure (rectangular parallelepiped and sphere). The results shown in Fig. 6 prove that the reconstruction accuracy decreases with the increase of noise level and the decrease of number of projections. However, regardless of the operator used (SF-TT, SDD or RD), the reconstruction accuracy does not vary. Anal- ysis of Fig. 6 enables us to divide the results of the reconstruction in three groups: a first group for which the NMSE is less than 0.07 ( SN RV AR ≥ 50db),where we can say that the reconstruction is good, a second group for which NMSE ∈ [0.07, 0.25] (20db ≤ SN RV AR < 50db), where we have a weak reconstruction and we can hope to improve these results with a simple contour regularization, and finally, the last group for which NMSE is greater than 0.25 (SN RV AR < 20) where the reconstruction is very bad, so the necessity of introducing high-level constraints.

Conclusion and perspective

We proposed a 3D object-based binary reconstruction without regularization based on a new simplified Distance-driven (SDD) projector and a level set approach which acts as an implicit regularizer.

In this experiment, we studied the reconstruction quality using SF-TT, RD and SDD projectors in the case of noisy and sparse data from cone-beam acquisitions. A binary Shepp-Logan phantom was used for performance evaluation in varying the number of projections and the gaussian noise level. Considering the limited number of projections, there is no difference in the reconstruction accuracy term and convergence speed between these three approaches. However, our reconstruction converge faster when using SDD projector compared with the SF-TT and RD operators, in the absence of noise.

The results show also a proportional degradation of the binary reconstruction with the increase of noise, and we have established preliminary classification on the limits of such reconstruction in presence of noise, and a limited number of projections. Future work will be oriented to find the best a priori, to compensate the degradation introduced by the noise, and to consider a gray-scale reconstruction.

Fig. 1 .

 1 Fig. 1. Acquisition parameters of axial 3D cone-beam flat detector geometry.

Fig. 2 .

 2 Fig. 2. Principle of the Separable Footprint method: Trapeze Trapeze SF-TT

Fig. 3 .

 3 Fig. 3. Principle of Simplified Distance-driven method.

  4.I). The digitization is performed in NX × NY × NZ = 128 × 128 × 128 voxels of size ∆X = ∆Y = ∆Z = 1.2mm. The flat-detector is simulated with a size of Ns × Nt = 128 × 128 cells spaced by ∆s = ∆t = 2mm, located at a distance D sd = 1103mm from the X-ray source.The whole "source-detector" rotates around a fixed Z-axis, such as Dso = 690mm.

Fig. 4 .

 4 Fig. 4. Projection results. (I): binary 3D digital phantom. (II): projection results for θ = 0 • , Line 1: using SF-TT matrix, Line 2: using SDD matrix, column A: without noise, column B: with noise SN RV AR=25 db, column C: with noise SN RV AR=20 db.

Fig. 5 .Fig. 6 .

 56 Fig. 5. NMSE according to iterations. line 1: without noise, line 2: withe noise SN RV AR=25 db, line 3: withe noise SN RV AR=20 db, column A: using 13 projections, column B: using 7 projections, column C: using 5 projections.

ACKNOWLEDGMENTS

This work was supported by ANR grant MESANGE (ANR-08-BLAN-0198), the National Natural Science Foundation (No. 31100713) and French National Institute for Health (INSERM).


Martina Cardone
email: cardone@eurecom.fr

Daniela Tuninetti
email: danielat@uic.edu

Raymond Knopp
email: knopp@eurecom.fr

Umer Salim
email: umer.salim@intel.com

The symmetric sum-capacity of the Gaussian Half-Duplex Causal Cognitive Interference Channel to within a constant gap

This paper studies the sum-capacity of the Gaussian half-duplex causal cognitive interference channel, a channel model with two transmitter-receiver pairs where a (cognitive) source cooperates with the other (primary) source in sending data through a shared channel. In contrast to the classical cognitive radio model, here the cognitive source can not transmit and receive at the same time and must causally learn the primary message through a noisy channel. Achievable strategies are developed and shown to match known upper bounds on the symmetric sum-capacity of this channel to within a constant gap for all values of channel parameters. In the process, the generalized degrees of freedom of the channel is characterized.

I. INTRODUCTION

We consider the Gaussian cognitive overlay channel characterized by two sources, namely PTx and CTx, and two destinations, namely PRx and CRx. Each source wishes to communicate with the corresponding receiver (PTx with PRx, CTx with CRx). PTx and PRx are deployed by the spectrum license owner and are referred to as the primary pair. CTx and CRx opportunistically use the licensed spectrum and are referred to as the cognitive pair. The classical information theoretic model for the cognitive overlay channel assumes that the CTx has a non-causal and perfect knowledge of the primary user's message prior to transmission [START_REF] Devroye | Achievable rates in cognitive radio channels[END_REF]. This assumption has been very useful to show the potential fundamental gains of a cognitive radio technology. However, from a practical point of view, it might not be realistic. In this work we remove this ideal assumption by imposing causality constraints for the CTx. To this end, the CTx must learn the primary message over a lossy communication link between PTx and CTx. Moreover, we assume that the CTx operates in Half-Duplex (HD) mode, i.e., it can either transmit or receive at any given time, but not both. We shall denote this system as the Half-Duplex Causal Cognitive Interference Channel (HD-CCIC).

Related Work. The exact characterization of the capacity region for a general memoryless network is challenging. Recently, it has been advocated that progress can be made towards determining the capacity by showing that achievable strategies are provably close to computable outer bounds [START_REF] Avestimehr | Wireless network information flow: a deterministic approach[END_REF]. For the non-causal cognitive channel in Gaussian noise the capacity region has been exactly determined for some parameter regimes and it is known to within 1 bit otherwise [START_REF] Rini | Inner and outer bounds for the gaussian cognitive interference channel and new capacity results[END_REF]. This work considers the HD-CCIC, which can be studied within the framework of the Interference Channel (IC) with generalized feedback, or with bilateral Full-Duplex (FD) source cooperation [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF], [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF], [START_REF] Yang | Interference channel with generalized feedback (a.k.a. with source cooperation): Part i: Achievable region[END_REF], by imposing that only the CTx cooperates and by using the formalism of [START_REF] Kramer | Models and theory for relay channels with receive constraints[END_REF]. In [START_REF] Kramer | Models and theory for relay channels with receive constraints[END_REF], it was showed that there is no need to develop a separate theory for memoryless networks with HD nodes as the HD constraints can be incorporated into the memoryless FD framework by introducing a "state random variable" in the input definition. In [START_REF] Kramer | Models and theory for relay channels with receive constraints[END_REF], it was also proved that the largest rate can be achieved by randomly switching between the transmit-and receive-phases at the HD relay node in a simple relay channel. In this way, the randomness that lies into the switch can be used to transmit (at most 1 bit per channel use of) information to the destination. We shall refer to this specific HD mode of operation as random switch [START_REF] Kramer | Models and theory for relay channels with receive constraints[END_REF], as opposed to deterministic switch where the transmit-and receive-phases are predetermined.

Outer and inner bounds for the sum-capacity of the Gaussian IC with either FD source or destination bilateral cooperation were derived in [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF]; in [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF], a sum-rate outer bound for the general memoryless IC with source cooperation was derived by extending the work of [START_REF] Kramer | Outer bounds on the capacity of gaussian interference channels[END_REF]; in [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF], a sum-rate outer-bound for a special class of "injective" IC with FD source cooperation was proposed by extending the work of [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF]. The largest known achievable region for the general memoryless IC with FD source cooperation, to the best of our knowledge, is in [START_REF] Yang | Interference channel with generalized feedback (a.k.a. with source cooperation): Part i: Achievable region[END_REF]. In [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF]Prop.3] the symmetric sum-capacity of the Gaussian IC with FD source cooperation was characterized to within 6 bits, and in [START_REF] Yang | Interference channels with source cooperation in the strong cooperation regime: Symmetric capacity to within 2 bits/s/hz with dirty paper coding[END_REF] to within 2 bits in the "strong cooperation regime".

The Gaussian IC with bilateral and unilateral HD source cooperation was studied in [START_REF] Wu | Interference Channels with Half-Duplex Source Cooperation[END_REF] where the sum-capacity was characterized to within 20 bits and 31 bits, respectively, by considering deterministic switch at the CTx in both inner and outer bounds. In this work we overcome a number of limitations and improve on the results of [START_REF] Wu | Interference Channels with Half-Duplex Source Cooperation[END_REF] as follows: (i) we properly account for random switch at the CTx in the outer bound, (ii) we consider the classical definition of sumcapacity without introducing any 'back-off' in the PTx rate, (iii) we derive the generalized Degrees-of-Freedom (gDoF) in closed form rather than expressing it implicitly as the solution of a linear program, and (iv) we reduce the gap to 10.503 bits. Contributions and Paper Outline. We focus on the symmetric Gaussian HD-CCIC, described in Section II, and characterize its gDoF in closed-form and its sum-capacity to within a constant gap in Section IV. In order to accomplish this we first specialize, in Section III, the sum-rate bounds of [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF], [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF], [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF] to the HD-CCIC by using the formalism of [START_REF] Kramer | Models and theory for relay channels with receive constraints[END_REF]. We then develop achievable schemes that are asymptotically optimal for different parameter regimes. These schemes could possibly be obtained as a special case of [START_REF] Yang | Interference channel with generalized feedback (a.k.a. with source cooperation): Part i: Achievable region[END_REF]; due to the complexity of [START_REF] Yang | Interference channel with generalized feedback (a.k.a. with source cooperation): Part i: Achievable region[END_REF] we develop schemes inspired by the linear deterministic approximation of the Gaussian HD-CCIC in the spirit of [START_REF] Avestimehr | Wireless network information flow: a deterministic approach[END_REF] (not reported here for space limitations but available at [START_REF] Cardone | On the gaussian interference channel with half-duplex causal cognition[END_REF]). An interesting feature of these schemes is that they employ successive decoding at the receivers, which is simpler than joint decoding. Moreover, our novel achievable schemes give insights on when causal cognitive radio might be of practical interest, i.e., when it outperforms the non-cooperative case, and it performs as bilateral cooperation.

PTx

CTx PRx CRx W p W c + + + Z c Z f Z p h pf h pp h pc h cp X p (X c , M c ) Y p Y c Y f h cc

II. SYSTEM MODEL

General Memoryless Channel Model. A CCIC is described by two input alphabets (X p , X c ), three output alphabets (Y f , Y p , Y c ) and a memoryless transition probability P Y f ,Yp,Yc|Xp,Xc . PTx aims to transmit a message W p ∈ [1 : 2 N Rp ] to PRx and CTx wishes to send a message W c ∈ [1 : 2 N Rc ] to CRx, where N denotes the codeword length and R p and R c are the transmission rates for PTx and CTx, respectively, measured in bits per channel use (logarithms are in base 2). The messages W p and W c are independent and uniformly distributed on their respective domains. At time i ∈ [1 : N ] PTx encodes its message W p into X p,i (W p ) and CTx maps its message W c and its past channel observations into X c,i (W c , Y i-1 f ). At time N , PRx estimates its intended message W p based on all its channel observations as W p (Y N p ), and similarly CRx outputs W c (Y N c ). The capacity region is defined as the convex closure of all non-negative rate pairs

(R p , R c ) such that max u∈{c,p} P[ W u = W u ] → 0 as N → +∞.
Gaussian HD-CCIC (G-HD-CCIC). HD channels represent a special case of the memoryless FD framework [START_REF] Kramer | Models and theory for relay channels with receive constraints[END_REF]. With a slight abuse of notation compared to the previous paragraph, let the channel input at the CTx be the pair (X c , M c ), where X c ∈ X c as before and M c ∈ {0, 1} is the state random variable that indicates whether the CTx is in receive-mode (M c = 0) or in transmit-mode (M c = 1). The single-antenna G-HD-CCIC, in Fig. 1, has the input/output relationship

  Y f Y p Y c   =   1-M c 0 0 0 1 0 0 0 1   H 1 0 0 M c X p X c +   Z f Z p Z c   , (1)
where the inputs are subject to the average power constraints

E |X i | 2 ≤ 1, i ∈ {p, c}
, and H is the constant complexvalued channel matrix that defines the connectivity of the network defined in the symmetric case as

H :=   h pf h pp h cp h pc h cc   =   √ C √ S √ Ie jθ1 √ Ie jθ2 √ S   , (2)
where indicates the channel gain that does not affect the capacity region (because CTx can remove its transmitted signal X c from its channel output Y f ) and where some channel gains can be taken to be real-valued and non-negative because a node can compensate for the phase of one of its channel gains. The noises are proper-complex Gaussian random variables with parameter Z k ∼ N (0, 1), k ∈ {f, p, c}. In the following we assume that the noises are independent. Performance metrics. The capacity of the channel in (1) is unknown. Here we make progress by first determining its gDoF and then showing that the sum-capacity outer-bound is achievable to within a constant gap for any value of the channel parameters. The gDoF of a symmetric Gaussian noise channel is defined as [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF] d(α, β) := lim

S→+∞ max{R p + R c } 2 log(1 + S) , (3)
where the maximization is over all achievable rate pairs (R c , R p ) and where, for some fixed non-negative (α, β), we parameterized C = S β and I = S α . Here S represents the SNR on the direct links.

The gDoF region gives an asymptotically exact characterization of the capacity at infinite SNR. At finite SNR the sum-capacity is said to be known to within GAP bits if we can show an inner bound (R p +R c ) (IB) and an outer bound

(R p +R c ) (OB) such that 0 ≤ (R p +R c ) (OB) -(R p +R c ) (IB) ≤ GAP
for all channel parameters (S, C, I, θ 1 , θ 2 ).

III. OUTER BOUNDS

In the literature several outer-bounds are known for bilateral source cooperation, where each source works in FD mode [START_REF] Host-Madsen | Capacity bounds for cooperative diversity[END_REF], [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF], [START_REF] Tuninetti | An outer bound region for interference channels with generalized feedback[END_REF]. Here we specialize them for the case of HD unilateral cooperation by following [START_REF] Kramer | Models and theory for relay channels with receive constraints[END_REF]. In particular, we proceed through the following steps: (i) in the outer bounds for the general memoryless channel with input X c , we substitute X c with the pair (X c , M c ); (ii) we apply the mutual information chain rule so as for any triplet of random variables (A, B, C) we have the bound

I(A, X c , M c ; B|C) ≤ H(M c ) + I(A, X c ; B|C, M c ); R p + R c ≤ min{(R p + R c ) (OB-CS) , (R p + R c ) (OB-DT) , (R p + R c ) (OB-PV) }, (4)
(R p + R c ) (OB-CS) := 2.507 + γ log (1 + S + C) + 2 (1 -γ) log (1 + S) , (5)
(R p + R c ) (OB-DT) := 3 + γ log (1 + S) + (1 -γ) log max {I, S} I + (1 -γ) log 1 + √ S + √ I 2 , (6)
(R p + R c ) (OB-PV) := 5.503 + γ log (1 + S + C + I) + 2 (1 -γ) log 1 + I + S + 2 √ IS 1 + I . ( 7
)
this follows since for a binary-valued random variable M c we have

I(M c ; B|C) ≤ H(M c ) = H(γ) for some γ := P[M c = 0] ∈ [0, 1]
and where H(γ) is the binary entropy defined as for ∈ {0, 1} and for some (P p,0 , P p,1 , P c,0 , P c,1 ) ∈ R 4 + satisfying the average power constraint

H(γ) = -γ log(γ) -(1 -γ) log(1 -γ); (iii)
γP u,0 +(1 -γ) P u,1 ≤ 1, u ∈ {p, c} ;
(iv) we rewrite the average power constraint as follows: since PTx transmits in both states M c = 0 and M c = 1 we define, for some β ∈ [0, 1], P p,0 = β γ and P p,1 = 1-β 1-γ ; since the CTx transmission only affects the receivers output when M c = 1, we let P c,0 = 0 and P c,1 = 1 1-γ ; (v) finally, the upper bounds in (5) (obtained from the cut-set bound [4, eq.(81)+eq.(82)]), in (6) (obtained from [8, eq.(4d)]) and in (7) (obtained from [5, eq.(6-7)]), given at the top of this page, are obtained by upper bounding each individual mutual information term over (ρ 0 , ρ 1 , β) ∈ [0, 1] 3 , for a fixed γ; (vi) after these steps, each outer bound has terms that are linear combinations of terms of the type 'γ log 1 γ '; by numerically optimizing these quantities, we obtain the constants that characterize the outer bounds in ( 5), ( 6) and [START_REF] Kramer | Models and theory for relay channels with receive constraints[END_REF].

IV. GDOF AND SUM-CAPACITY TO WITHIN A CONSTANT

Our first main results is:

Theorem 1 The gDoF of the symmetric G-HD-CCIC is d(α,β) ≤ 1 2 max γ min {γ max {1, β} + 2 (1 -γ) , (8a)
γ + (1 -γ) max {1, α} + [1 -α] + , (8b)
γ max {α, β, 1}+(1-γ) max {2α, 2-2α}} (8c) =                1-α+ 1 2 [β-2+2α] + α β+α-1 α ∈ [0, 1/2) α+ 1 2 [β-2α] + (2-3α) β-3α+1 α ∈ [1/2, 2/3) 1 -1 2 α α ∈ [2/3, 1) 1 2 α α ∈ [1, 2) 1 + 1 2 [β-2] + (α-2) β+α-3 α ∈ [2, ∞). (8d)
Proof: The proof that (8d) upper bounds the gDoF follows by substituting the sum-rate upper bounds in ( 5), ( 6) and ( 7) into the gDoF definition in [START_REF] Rini | Inner and outer bounds for the gaussian cognitive interference channel and new capacity results[END_REF]. The proof that (8d) is achievable is given in the Appendix.

Fig. 2 shows the optimal gDoF and the gap for the symmetric G-HD-CCIC. The whole set of parameters has been partitioned into multiple sub-regions depending upon different levels of cooperation (β) and interference (α) strengths.

The gDoF in (8d), in the different interference regimes (i.e., different values of α), is the sum of the gDoF without cooperation (obtained for β = 0) and a term that is strictly positive only for sufficiently large β. In particular, β must be larger than the gDoF without cooperation. This can be explained as follows:

• In weak interference α ∈ [0, 2/3) and without cooperation β = 0 the PV bound in (8c) is the tightest [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF]. The PV bound in (8c) is optimized by γ = 0, which is equivalent to nocooperation, whenever max{1, α, β} ≤ 2 max{1 -α, α} = 2d (NoCoop) (α) ⇐⇒ β ≤ 2d (NoCoop) (α). Thus, unilateral HD cooperation is beneficial in terms of gDoF only when β is larger than the gDoF without cooperation. In this case, the optimal γ, obtained by equating the PV bound in (8c) to the DT bound in (8b), is

γ (PV.DT) = min{2 -3α, α} min{2 -3α, α} + β -1 , (9)
which then leads to the first two cases in (8d).

• In very strong interference α ≥ 2 and without cooperation β = 0 the CS bound in (8a) is the tightest [START_REF] Kramer | Outer bounds on the capacity of gaussian interference channels[END_REF]. The CS bound in (8a) is optimized by γ = 0, which is equivalent to nocooperation, whenever max{1,β} ≤ 2 = 2d (NoCoop) (α) ⇐⇒ β ≤ 2d (NoCoop) (α). Again we see that unilateral HD cooperation is beneficial in terms of gDoF only when β is larger than the gDoF without cooperation. In this case, the optimal γ, obtained by equating the CS bound in (8c) to the DT one in (8b), is

γ (CS.DT) = α -2 β + α -3 , (10)
which then leads to the last case in (8d).

• In moderately-weak and strong interference α ∈ [2/3, 2) and with no cooperation β = 0 the DT bound in (8b) is the tightest [START_REF] Kramer | Outer bounds on the capacity of gaussian interference channels[END_REF]. This bound is always optimized by γ = 0, which is equivalent to no-cooperation. Hence in this regime it is always gDoF-optimal to operate the channel as a pure IC. Summarizing the above results we have: • In regions 1, 2, 4, 5, 6, 7 and 9 in Fig. 2, unilateral HD cooperation might not be worth implementing in practical systems since the same gDoF is achieved without cooperation.

` !! !! 1 2 ! 2 3 ! 1! 2! 1! 2! 1" 3" 2" 4" 5" d(!, !) = 1! !!!!!!!!GAP!=!1!bit! ! d(!, !) = 1! GAP!=!2.507!bits! d(!, !) = 1 2 !! -2 ! + ! -3 ! !!!!!!!!!!!!!!!!!!!!!!!!!GAP!=!5!bits! d(!, !) = ! 2 ! !!GAP!=!1!bit! d(!, !) = 1 - ! 2 ! GAP!=!3!bits! ! = 2!! 6" 8" d(!, !) = !! GAP!=!7.322!bits! d(!, !) = 1 2 2! -!" -2! ! -3! + 1 ! GAP!=!10.503!bits! ! = 2 -2!! 10" 9" d(!, !) = 1 -!! !!!!!GAP!=!8.673!bits! d(!, !) = 1 2 2! + 2! -!" -2 ! + ! -1 ! GAP!=!9.503!bits! 7" d(!, !) = !! GAP!=!9.503!bits! ! = 2! -1!
• In regions 1, 4, 5 and 6 the gDoF is equal to the one with unilateral FD cooperation [START_REF] Cardone | On the gaussian interference channel with causal cognition, or with unilateral source cooperation[END_REF]. This implies that, in these regimes, the same gap results found for the FD case in [START_REF] Cardone | On the gaussian interference channel with causal cognition, or with unilateral source cooperation[END_REF] hold in the case of unilateral HD cooperation. In these regimes there is no penalty in HD cooperation.

• By computing the gDoF of the IC with bilateral HD source cooperation [START_REF] Cardone | Approximate sumcapacity of full-and half-duplex asymmetric interference channels with unilateral source cooperation[END_REF] we see that in regions 1, 2, 6, 7, 9, part of 5 (with β ≤ 3 -2α) and part of 4 (with β ≤ 2α -1), the gDoF are the same as those computed in the unilateral case. Thus, for this set of parameters, unilateral cooperation attains the same gDoF of bilateral cooperation but with less resources and therefore represents a better trade-off in practical systems.

Our second main result is:

Theorem 2 The sum-rate upper bound in (4) is achievable to within 10.503 bits for the symmetric G-HD-CCIC.

Proof: The proof can be found in the Appendix. Before concluding we make some more remarks on the gap: • Most of our proposed schemes only use superposition coding. The only exception is in weak interference, i.e., regions 8 and 10 in Fig. 2, where we were not able to characterize the sum-rate to within a constant gap with superposition only.

• All the achievable schemes presented here use successive decoding at the receivers. This form of decoding, in practice, is simpler than joint decoding; thus our schemes may be used as guidelines to deploy practical cognitive radio systems.

• The gap computed in this work is around 3 bits bigger than the one computed in the FD case [START_REF] Cardone | On the gaussian interference channel with causal cognition, or with unilateral source cooperation[END_REF]. Possible ways to reduce this gap may be: (i) apply joint decoding at the receivers, (ii) develop block Markov schemes instead of taking inspiration by the linear deterministic approximation channel of the Gaussian noise channel at high SNR, (iii) develop achievable schemes that exploit the randomness in the switch to convey further useful information, (iv) develop tighter upper bounds than those used in this work, especially in weak interference where the gap is quite large.

V. CONCLUSIONS

In this work we considered the half-duplex causal cognitive channel. Differently from the original overlay cognitive model, in which the CTx has a full a priori knowledge of the message of the PTx, we consider a more practical scenario in which the CTx learns this message through a noisy link and

(Rp + Rc) (IB 1) =γ * log (1 + S) -γ * log 1 + S 1 + S + γ * log 1 + C 1 + S + (1 -γ * ) log (1 + S) + (1 -γ * ) log (1 + S) -(1 -γ * ) log 1 + S 1 + S , (Rp + Rc) (IB 2) =γ * log (1 + S) -γ * log 1 + S 1 + S + γ * log 1 + C 1 + S + 2 (1 -γ * ) log 1 + I + S 1 + I -2 (1 -γ * ) log 1 + I 1 + I + (1 -γ * ) log 1 + S 1 + I + I 1 + I -(1 -γ * ) log 1 + SI + I + I 2 (1 + I) 2 + S 1 + I , (Rp + Rc) (IB 3) =γ * log (1+S) -γ * log 1+ S 1 + S + γ * log 1+ C 1 + S -(1 -γ * ) log (1+I) + (1 -γ * ) log 1+I+ S 1 + I + (1 -γ * ) log 1+ I 1 + I + S -(1 -γ * ) log 1+ I 1 + I .
operates in HD mode. We derived the gDoF of the channel and characterized its symmetric sum-capacity to within a constant gap. This has been accomplished by developing novel achievable schemes for the different parameter regimes.

ACKNOWLEDGMENT

The work of D. Tuninetti was partially funded by NSF under award number 0643954 and thanks to the generous support of Telecom-ParisTech, Paris France, while the author was on a sabbatical leave at the same institution. Eurecom's research is partially supported by its industrial partners: BMW Group Research & Technology, IABG, Monaco Telecom, Orange, SAP, SFR, ST Microelectronics, Swisscom and Symantec.The research carried out at Eurecom leading to these results has received funding from the EU Celtic+ Framework Program Project SPECTRA. The research at IMC leading to these results has received funding from the EU FP7 grant agreement iJOIN n • 317941.

APPENDIX

We develop achievable schemes inspired by the linear deterministic approximation of the G-HD-CCIC at high SNR. The details of the derivation may be found in [START_REF] Cardone | On the gaussian interference channel with half-duplex causal cognition[END_REF].

• Region 1 (α ≥ 2 and β ≤ 1), Region 2 (α ≥ 2 and 1 < β ≤ 2), Region 4 (1 ≤ α < 2), Region 5 ( 2 3 ≤ α < 1), Region 6 ( 1 2 ≤ α < 2 3 and β ≤ 2α-1), Region 7 ( 1 2 ≤ α < 2 3 and 2α-1 < β ≤ 2α) and Region 9 (α < 1 2 and β ≤ 2-2α). In these regions the gDoF upper bound equals that of the classical IC. Therefore an achievable scheme is given by the classical IC. The gap is at most of GAP ≤ 9.503 bits.

• Region 3: α ≥ 2 and β > 2. With the achievable scheme in [13, Appendices D-A, D-B], i.e., a two-phase strategy with superposition coding and successive decoding, we can achieve (R p +R c ) (IB 1) at the top of the page, where the limit behavior of γ * is the one in [START_REF] Etkin | Gaussian interference channel capacity to within one bit[END_REF]. The tightest upper bounds are those in [START_REF] Prabhakaran | Interference channels with source cooperation[END_REF] and in [START_REF] Yang | Interference channel with generalized feedback (a.k.a. with source cooperation): Part i: Achievable region[END_REF]. The gap is GAP ≤ 5 bits.

• Region 8: 1 2 ≤ α < 2 3 and β > 2α. With the achievable scheme in [13, Appendices D-A, D-C], i.e., a two-phase strategy with superposition coding, dirty-paper-coding and successive decoding, we can achieve (R p +R c ) (IB 2) at the top of the page, where the limit behavior of γ * is the one in [START_REF] Kramer | Outer bounds on the capacity of gaussian interference channels[END_REF]. The tightest upper bounds are those in [START_REF] Yang | Interference channel with generalized feedback (a.k.a. with source cooperation): Part i: Achievable region[END_REF] and in [START_REF] Kramer | Models and theory for relay channels with receive constraints[END_REF]. The gap is GAP ≤ 10.503 bits.

• Region 10: α < 1 2 and β > 2 -2α. With the achievable scheme in [13, Appendices D-A, D-D], a two-phase strategy with superposition coding, dirty-paper-coding and successive decoding, we can achieve (R p +R c ) (IB 3) at the top of the page, where the limit behavior of γ * is the one in [START_REF] Kramer | Outer bounds on the capacity of gaussian interference channels[END_REF]. The tightest upper bounds are those in [START_REF] Yang | Interference channel with generalized feedback (a.k.a. with source cooperation): Part i: Achievable region[END_REF] and in [START_REF] Kramer | Models and theory for relay channels with receive constraints[END_REF]. The gap is GAP ≤ 9.503 bits.

Fig. 1 .

 1 Fig. 1. The 2-user G-HD-CCIC.

Fig. 2 .

 2 Fig. 2. Optimal gDoF and constant gap for the symmetric G-HD-CCIC in the different regimes of (α, β).

  for all the remaining mutual information terms, which are conditioned on M c , the 'Gaussian maximizes entropy'-principle [14, Appendix 16A] guarantees that we can restrict attention to joint Gaussian inputs with covariance matrix

	Cov	X p X c Mc=	=	P p, ρ * P p, P c,	ρ	P p, P c, P c,	: |ρ | ≤ 1