L Fatibene

M Ferraris

M Francaviglia

Extended Loop Quantum Gravity *

We discuss constraint structure of extended theories of gravitation (also known as f (R) theories) in the vacuum selfdual formulation introduced in [1].

Introduction

We have recently investigated a formulation of f (R) theories (in a metric-affine framework) based on non-linear actions similar to the Holst Lagrangian; see [START_REF] Fatibene | New Cases of Universality Theorem for Gravitational Theories[END_REF]. These actions are in fact written in terms of the scalar curvature β R of the Barbero-Immirzi connection with parameter β (see [START_REF] Barbero | Real Ashtekar variables for Lorentzian signature space-time[END_REF], [START_REF] Immirzi | Quantum Gravity and Regge Calculus[END_REF]) and are dynamically equivalent to the corresponding "classical" f (R) theory. For the linear case f ( β R) = β R one obtains the standard Holst action. Hence these new actions are to be understood as Barbero-Immirzi formulations of the corresponding classical f (R) theory. This could be interesting for at least two reasons: from the point of view of LQG this new formulation provides a family of models which are classically well-understood and investigated in detail (see [START_REF] Capozziello | Cosmography of f (R) gravity[END_REF], [START_REF] Nojiri | Modified gravity as realistic candidate for dark energy, inflation and dark matter[END_REF]). There are many classical effects known in f (R) theories that should be traced in their quantum genesis. The minisuperspace of these models is quite well-understood and should be studied in loop quantum cosmology (LQC) formulation (see [START_REF] Bojowald | Consistent Loop Quantum Cosmology Class[END_REF]), to contribute to a better understanding of the classical limit of LQG models. Moreover, as in all metricaffine models, matter has a non-trivial feedback on the gravitational field which would be also interesting to trace in its quantum origin. It is often said that matter in LQG simply adds new labels to spin networks, while in these models one could expect a more complicated mechanism that would be certainly interesting to be discussed in detail. Finally, there are a number of equivalences, e.g. with scalar tensor models (see [START_REF] Magnano | Nonlinear gravitational Lagrangians[END_REF]), that again would be interesting to be discussed in detail at quantum level. Let us stress that these equivalences are known to hold at the classical level and, as usual, one should investigate whether they still hold at the full quantum level or just emerge classically.

From the classical viewpoint we shall here provide a route to define a quantization à la loop of f (R) theories. Of course classical effects of these extended theories of gravitation have been extensively investigated. It is therefore interesting to investigate also their quantum effects. For example it would be interesting to see whether the removal of singularities that has been shown to hold in standard loop quantization of GR is preserved generically in these extended gravitational models.

For the sake of simplicity we shall here restrict our attention to the Euclidean signature and to the selfdual formulation (which in the Euclidean sector is in fact a special case of the Barbero-Immirzi formulation) and show that one can apply LQG methods (see [START_REF] Rovelli | Quantum Gravity[END_REF]) also to the quantization of these theories. In vacuum we shall obtain something similar to Einstein gravity with a cosmological constant. This is very well expected on the basis of a classical equivalence (see [START_REF] Borowiec | Universality of Einstein Equations for the Ricci Squared Lagrangians[END_REF]); however, let us stress that our result seems to establish a stronger equivalence at the quantum level and not only at the classical level.

Moreover, let us stress that the classical equivalence holds only in vacuum, while the equivalence is broken when generic matter is considered and the extended models are equivalent to scalar tensor theories; see [START_REF] Magnano | Nonlinear gravitational Lagrangians[END_REF]. Tracing the mechanism which leads to this shift of equivalence at the quantum level would be therefore rather interesting and will be investigated in forthcoming papers. We shall follow the notation introduced in [1] and [START_REF] Rovelli | Quantum Gravity[END_REF].

The aim of this paper is to go towards a quantum description of f (R) theories; we have however to mention the reverse problem of giving a (semi)-classical account of the quantum effect of ordinary standard LQG models; see [START_REF] Olmo | Covariant Effective Action for Loop Quantum Cosmology a la Palatini[END_REF]. The two approaches are somehow complementary and based on similar techniques.

Selfdual Formulation for Extended Theories

In where e µ a is a spin frame (see [START_REF] Fatibene | Natural and gauge natural formalism for classical field theories. A geometric perspective including spinors and gauge theories[END_REF]), R ab µν is the curvature of a spin connection ω ab µ on a 4 dimensional (spin) manifold M and β = 0 is a real parameter. Indices a, c, . . . = 0..3 and µ, ν, . . . = 0..3 while i, j, . . . = 1..3.

In the Euclidean sector one obtains for β = 1 2 the standard selfdual curvature which can be written in terms of the curvature F i µν := p i ab R ab µν of the usual selfdual connection A i µ := p i ab ω ab µ = ω 0i µ + 1 2 i jk ω jk µ as follows

1 2 + R = 1 2 R cd µν δ a [c δ b d] + 1 2 cd ab e µ a e ν b = R cd µν p i cd p ab i e µ a e ν b = p ab i F i µν e µ a e ν b =: F (2.

3)

Here p ab i denotes the algebra projector p : spin(4) → su(2) on selfdual forms. It is given by

p 0j i = 1 2 δ j i p j0 i = -p 0j i p jk i = 1 2 i jk (2.4)
and the inverse projector p i ab is defined by

p i 0j = 1 2 δ i j p i j0 = -p i 0j p i jk = 1 2 i jk (2.5)
One can easily prove that

p i ab p ab j = δ i j p i ab p cd i = 1 2 δ a [c δ b d] + 1 2 ab cd (2.6)
One is then led to consider the following family of Lagrangians

L + = 1 2κ ef (F ) + L m (2.7)
where κ = 8πG, e is the determinant of the frame matrix, f is a generic analytic function and L m encodes the matter contribution. Usually matter is assumed to couple only with g (and possibly to its derivatives up to some finite order; usually, in view of minimal coupling principle, at most 1) and not to the connection ω ab µ . Hereafter we shall just consider the vacuum sector, i.e. we set L m = 0.

In the special case f (F ) = F one obtains an equivalent formulation of the usual selfdual action

L + = 1 8κ + R ab ∧ e c ∧ e d abcd = 1 16κ R ab µν + 1 2 ab ef R ef µν e c ρ e d σ µνρσ abcd ds = = 1 14κ R ef µν δ a [e δ b f ] + 1 2 ab ef e c ρ e d σ µνρσ abcd ds = 1 8κ R ef µν p ab i p i ef e c ρ e d σ µνρσ abcd ds = = 1 8κ p ab i F i µν e c ρ e d σ µνρσ abcd ds = e 2κ p ab i F i µν e µ a e ν b ds = e 2κ F ds (2.8)
where ds is the standard local basis of 4-forms on M induced by coordinates.

Field equations of the Lagrangian L + are

p ab i F i µν e c ρ µνρσ abcd = 0 p i ab ∇ µ e a ν e b ρ µνρσ = 0 (2.9)
Let us now consider a Cauchy (boundary

) surface i : S → M : k A → x µ (k), A, B, . . . = 1..3; in coordinates x µ = (t, k A ) adapted to the submanifold S one has i : k A → k A and ∂ A x µ = δ µ A .
The unit covector normal to S is given by n = dx 0 . One can use antiselfdual transformations to define a canonical adapted frame e a = e

A i A = A i µ ∂ A x µ on S. Let us denote by F i AB = F i µν ∂ A x µ ∂ B x ν the
projected curvature (which is the same as the curvature of the projected connection); for later convenience let us also define the tangent-normal projection of the curvature F i A = F i µν ∂ A x µ n ν (of course the normal-normal projection vanishes due to the skew symmetry of F ).

Let us also set E A i = A i for the momentum conjugated to the connection A i A written in terms of the triad A i tangent to S, with the determinant of the (co)triad i A .

Field equations (2.9) can be projected onto S to obtain a number of evolution equations and the following constraints on S:

       A ∇ A E A i = 0 F i AB E A i = 0 i jk F i AB E A j E B k = 0 (2.11)
These constitute the starting point of LQG quantization scheme; the first equation is related to gauge covariance, the second to Diff(S)-covariance; while the third equation is called the Hamiltonian constraint, when quantized it becomes the so-called Wheeler-deWitt equation and it encodes the (quantum) dynamics. In order to solve the first and second equation one defines an Hilbert space spanned by spin knots (see [START_REF] Rovelli | Quantum Gravity[END_REF]) so that the Wheeler-deWitt equation is implemented as an operator on that space and it defines physical states.

On this basis one expects to be able to perform the same steps with extended models f (F ); since the extended models are still gauge and generally covariant, the first and second equations are expected to remain unchanged. This would mean that the definition of Area and Volume operators are unchanged and "spacetime" gets discretized in extended models exactly as in standard LQG. Since extended models are known to provide a modified dynamics with respect to standard GR one also expect that the Wheeler-deWitt equation has to be modified.

We shall hereafter compute the analogous of equations (2.11) for the action (2.7) in order to fully confirm our expectations.

Constraint Structure

Let us then consider the Lagrangian

L + = e 2κ f (F ) ( 3 .1)
i.e. the purely gravitational part of (2.7).

Field equations are

f p ab i F i µν e µ a -1 2 fe b ν = 0 p ab i ∇ µ (ef e µ a e ν b ) = 0 (3.2)
The master equation f F -2f = 0 is obtained by tracing the first one by means of e ν b ; see [1] and [9]. This can be replaced back into the first equation to obtain

f p ab i F i µν e µ a -1 4 F e b ν = 0 ⇒ p ab i F i µν e µ a -1 4 F e b ν = 0 (3.3)
where we used the fact that generically f = 0 on the zeroes of the master equation. For simplicity let us assume that the master equation has only one (simple) zero F = ρ; when there are many (simple) zeroes each of them defines a sector of the quantum theory and one is supposed to sum over all sectors, which are in correspondence with the discrete zero structure of the analytic function f .

Let us also define a conformal tetrad ẽa µ = |f |e a µ , set σ = sgn(f (ρ)) and use tilde to denote quantities depending on the conformal tetrad, e.g. ẼA

i = ˜ ˜ A i = |f |E A i and F = p ab i F i µν ẽµ a ẽν b = σ f F (3.4)
Field equations are hence equivalent to

       p ab i F i µν ẽµ a -1 4 F ẽb ν = 0 f F -2f = 0 ⇒ F = ρ p ab i ∇ µ (ẽẽ µ a ẽν b ) = 0 (3.5)
The third equation implies the constraint

A ∇ A ẼA i = 0 (3.6)
as in the standard case, though for the conformal frame ẽa µ . The second equation can be now expanded as

F = p ab i F i µν ẽµ a ẽν b = 2p 0l i F i µν ẽµ 0 ẽν l + p lk i F i µν ẽµ l ẽν k = -F i A ˜ A i + 1 2 i lk F i AB ˜ A l ˜ B k = σ f ρ (3.7)
which allows us to express F i A ˜ A i as a function of constrained fields, i.e.

F i A ˜ A i = 1 2 i lk F i AB ˜ A l ˜ B k -σ f ρ (3.8)
Notice that the first equation is really different from the standard case (i.e. LQG without cosmological constant) due to the different coefficient 1 4 (which in the standard case is 1 2 and allows a complete cancellation of F i A ˜ A i ). The standard case in LQG can be recovered by setting f (F ) = F ; in this case the master equation simply implies F = 0 and the standard case without cosmological constant is obtained in particular. The first equation can be projected in the normal direction to the constraint to obtain

p ab i F i µν ẽµ a -1 4 F ẽb ν ẽα b ẽν d n α = 0 ⇒ (3.9)
p j0 i F i µν ẽµ j ẽν d -1 4 F δ 0 d = 0 ⇒ (3.10) F i Aν ẽA i ẽν d + 1 2 F δ 0 d = 0 (3.11) For d = k = 1..3 one has F i AB ẽA i ẽB k = 0 ⇒ F i AB ẼA i = 0 (3.12)
For d = 0 one has instead

F i A ẽA i + 1 2 F = 0 (3.13)
and, using (3.7) and (3.8), one obtains

F i A ẽA i -1 2 F i A ˜ A i + 1 4 i lk F i AB ˜ A l ˜ B k = 1 2 F i A ˜ A i + 1 4 i lk F i AB ˜ A l ˜ B k = = 1 4 i lk F i AB ˜ A l ˜ B k -σ 2f ρ + 1 4 i lk F i AB ˜ A l ˜ B k = 1 2 i lk F i AB ˜ A l ˜ B k -σ 2f ρ = 0 (3.14) i lk F i AB ˜ A l ˜ B k = σ f ρ (3.15) i lk F i AB ẼA l ẼB k = σ f ρ˜ 2 = σ f ρ Ẽ (3.16)
where Ẽ := det(˜ ˜ A i ) = ˜ 3 ˜ -1 = ˜ 2 denotes the determinant of the conformal momentum ẼA i . Let us stress that all this can be done also in the standard LQG framework, though in that case F i A does not enter other constraints and hence can be ignored. Accordingly, the constraints can be written in terms of the conformal triad as follows

         A ∇ A ẼA i = 0 F i AB ẼA i = 0 i jk F i AB ẼA j ẼB k = σ f ρ Ẽ (3.17)
As expected, the first and second constraints are unchanged with respect to (2.11), while the Wheeler-deWitt equation is modified by the "source term" σ f ρ Ẽ, which explicitly depends on the non-linearity of f (F ). This is the quantum counterpart of what happens classically for f (R) theories and reflects also what happens in standard LQG with the cosmological constant Λ = -1 4|f | ρ; see Appendix A. Let us also notice that the third constraint is a density, which is fundamental in the approach to quantization proposed by Thiemann; see [START_REF] Thiemann | LoopQuantumGravity: An InsideView[END_REF].

Conclusions and Perspectives

We have shown that, in the generic extended models introduced in [1], constraints allows a loop theory approach to quantization formally similar to what one usually does in vacuum models with cosmological constant. This shows that the equivalence between f (R) models and Einstein with cosmological constant (shown in [START_REF] Borowiec | Universality of Einstein Equations for the Ricci Squared Lagrangians[END_REF] to hold in the classical theory) holds also at the quantum level.

Of course more attention should be paid when matter couplings are considered, when this equivalence is known to break and is replaced at least by a conformal equivalence. Also the whole Hamiltonian structure of the theory should be verified in detail to exclude second class constraints which might add further equations to the set (3.17). These constraints (3.17) are in any case necessary conditions on the boundary S. Since from them discretization of "spacetime" follows one can claim in any event that extended spacetimes are discretized as in standard LQG.

Appendix A. LQG with Cosmological Constant

Let us here briefly review the standard result for LQG quantization in vacuum with cosmological constant in order to compare it with what we found for extended models. which can also be written in terms of the selfdual curvature as

Let us consider the Lagrangian

L Λ = + R ab + Λ 6 e a ∧
L Λ = 2p ab i F i + Λ 6 e a ∧
      A ∇ A E A i = 0 F i AB E A i = 0 i jk F i AB E A i E B j = -4ΛE (A.4)
which account for the value of the cosmological constant as claimed after (3.17) in which, however, the conformal frame was used.

Appendix B. Matter Fields

We want here to briefly comment about some cases of models with matter. The general treatment of matter is quite difficult and it deserves further investigation. Matter fields in this context are complicated by two different reasons: first, in classical f (R) theories generic matter modifies the master equations to become f R -f = T where T is the g-trace of energymomentum tensor of matter T µν (i.e. Hilbert stress tensor). This generically still allows to determine R (and thence f (R)) as a function of matter fields. Accordingly, these quantities generically are not constant any longer, and they depend on the spacetime point x through the matter fields. Of course, there are special cases in which the energy-momentum tensor happens to be traceless which still can be treated easily, essentially as in the vacuum case.

Second, in LQG matter can be easily considered though the method is based on regarding it in terms of groups so that matter contribution can be suitably encoded in terms of holonomies. This is trivial for Yang-Mills fields and easy in a number of relevant examples; see [START_REF] Rovelli | Quantum Gravity[END_REF].

The connection A i µ in LQG is a SU(2)-connection, as described in detail in [14], [START_REF] Fatibene | Spacetime Lagrangian Formulation of Barbero-Immirzi[END_REF]. Hence it is a principal connection on a suitable SU(2)-bundle + P over the spacetime M . If one couples with a Yang-Mills matter field A I µ a new gauge group G is introduced and the gauge field is a principal connection over a bundle P with structure group G. Here the Lie algebra g of the (semisimple) Lie group G is of dimension n and T I denotes an orthonormal basis with respect to the Cartan-Killing metric on G.

The gauge field strength is assumed to be denoted as usual by F I µν and the Yang-Mills Lagrangian is

L m = -1 4 √ gδ IJ F I µν F J ρσ g µρ g νσ ds (B.1)
and as usual we shall set F µν I := δ IJ F J ρσ g µρ g νσ , Greek world indices are lowered and raised by the associated metric g while Latin algebra indices are lowered and raised by the Cartan-Killing metric δ on G.

The energy-momentum tensor is

T µν = F I µ α • F I να -1 4 F I αβ F αβ I g µν (B.2)
which when the spacetime is in dimension dim(M ) = 4 is in fact traceless.

The master equation is then exactly the same as in the vacuum case, and the conformal tetrad ẽa µ is defined exactly as in the vacuum case. Field equations are then in the form

             p ab i F i µν ẽµ a -1 4 F ẽb ν = κT µν ẽbµ f F -2f = 0 ⇒ F = ρ p ab i ∇ µ (ẽẽ µ a ẽν b ) = 0 ⇒ A ∇ A E A i = 0 ∇ µ ( √ gF µν I ) = 0 (B.3)
where, of course, now ∇ µ takes care, when necessary, also of the G-gauge transformations, besides spacetime diffeomorphisms.

For the sake of simplicity let us consider hereafter the case of electromagnetism, i.e. taking G = U (1) which being of dimension 1 leads to a systematic understanding of the algebra indices. These field equations can be shown to project on the Cauchy surface S to get the following equations

             A ∇ A ẼA i = 0 F i AB ẼA i = κ σ f B × E i jk F i AB ẼA i ẼB j = -4Λ Ẽ + κ σ f |E| 2 + |B| 2 ∇ A E A = 0 (B.4)
where we defined E A := F µν ∂ A x µ n ν and B A := 1 2 A BC F µν ∂ B x µ ∂ C x ν for the electric and magnetic field. These are the Hamiltonian constraints (together with ∇ A B A = 0) and they are the starting point of LQG quatization with matter coupling. The system is described in terms of a connection of the group SU(2) × U (1) and thence in terms of its holonomies. This should lead to define spin networks with extra label of irreducible representation of U (1). See [8] and references quoted therein.

Once again the conformal frame plays a preferred role. Except for that the model is equivalent to standard GR, with cosmological constant, coupled with electromagnetic field though with a modified coupling constant.

  [1] we introduced β R := R ab µν e µ a e ν b + βR ab µν e cµ e dν cdab (2.1)

µa

  ∂ µ and coframe e a = e a µ dx µ (see[START_REF] Fatibene | Spin Structures on Manifolds and Ashtekar Variables[END_REF]) given by  better spin frames; [11]) adapted to S define triads i = e i = α A i ∂ A on S. Also the selfdual connection can be projected onto S to define a connection

  e b ∧ e c ∧ e d abcd = ds = 2e + R ab µν e µ a e ν b + 2Λ ds (A.1)

e b ∧ e c ∧ e d abcd (A. 2 ) 3 )

 23 By varying this Lagrangian one gets the following field equations By projecting on the boundary S one gets the following constraints 

Acknowledgments

We wish to thank C. Rovelli for discussions about Barbero-Immirzi formulation. We acknowledge the contribution of INFN (Iniziativa Specifica NA12) and the local research project Leggi di conservazione in teorie della gravitazione classiche e quantistiche (2010) of Dipartimento di Matematica of University of Torino (Italy).


Darine Kaddour

Jean-Daniel Arnould

Philippe Ferrari

Miniaturized Semi-Lumped UWB Bandpass Filter with Improved out-of-band performances

A miniaturized ultra wideband (UWB) bandpass filter with improved out-of band performances is presented in this paper. Based on the combination of low-pass and high-pass filters, the UWB filter is realized in a semilumped technology using microstrip transmission lines and surface mounted capacitors. In this paper, the filter design rules have been carried out. Furthermore, filters having a 3-dB fractional bandwidth of 142% centered at 0.77 GHz have been realized. Measurements in good agreement with simulations, show attractive properties of return loss (|S 11 |<-18 dB), insertion loss (<0.3 dB). Moreover, a technique adopting capacitively loaded stubs was used to extend the stopband till 8 times the center frequency.

I. Introduction

Since the Federal Communications Commission (FCC)'s release of the frequency band from 3.1 to 10.6 GHz for commercial purposes [START_REF]Revision of Part 15 of the commission's Rules regarding Ultra-Wide-Band Transmission System[END_REF], the UWB radio system has been receiving great attention from both academy and industry. As an essential component, UWB bandpass filters with good performances, compact size and easilyimplemented structures are in high demand. So far, several prototypes of UWB filters have been reported.

In [START_REF] Ishika | A design of tunable UWB filters[END_REF], a microstrip ring filter with the dual stopbands below 3.1 GHz and above 10.6 GHz was constructed to make up the most initial UWB filter. However, this filter in fact has many problematic issues, such as unexpected passbands below 3.1 GHz, narrow stopbands, large size and complex configuration.

Due to their geometric simplicity, parallel coupled microstrip lines were commonly used to design bandpass filters. Generally, this procedure is used for bandwidths less than about 20% [START_REF] Chin | New formulas for synthesizing microstrip bandpass filters with relatively wide bandwidths[END_REF]. To overcome this limitation, a three-coupled-line microstrip structure was proposed in [START_REF] Kuo | Wideband bandpass filter design with three-line microstrip structures[END_REF] to design a bandpass filter with a fractional bandwidth of 50 %. However, the more the fractional bandwidth is required, the smaller the gap size is demanded to enhance the coupling. For UWB filter design, the necessary gap size is still too narrow to be fabricated. Furthermore, ground plane aperture compensation techniques were successfully used for significant increasing of coupling coefficient between parallel coupled lines leading thus to wider bandwidths [START_REF] Kolmakov | Compact ultra-wideband bandpass filter with defected ground plane[END_REF][START_REF] Mandal | Compact wideband bandpass filter[END_REF].

Moreover, the stepped-impedance resonators (SIRs) demonstrated their advantageous efficiency for UWB bandpass filters. In [START_REF] Zhu | Ultra-wideband (UWB) bandpass filters using multiple-mode resonator[END_REF], a UWB bandpass filter with five transmission poles is proposed using multiple-mode resonator (MMR). By properly allocating the first three resonance peaks quasi-equally within UWB band, a short-circuited coplanar waveguide (CPW) MMR bandpass filter is realized [START_REF] Gao | Short-circuited CPW multiple-mode resonator for ultra-wideband (UWB) bandpass filter[END_REF]. In [START_REF] Sun | Capacitive-ended interdigital coupled lines for UWB bandpass filters with improved out-of-band performances[END_REF], a two-stage UWB bandpass filter is presented. The capacitive-ended interdigital coupled-line topology is adopted to achieve improved out-of-band per-formance. Note that the filters in [START_REF] Zhu | Ultra-wideband (UWB) bandpass filters using multiple-mode resonator[END_REF] and [START_REF] Sun | Capacitive-ended interdigital coupled lines for UWB bandpass filters with improved out-of-band performances[END_REF] lack of design method and synthesis procedure. Later, modified stubloaded multiple-mode resonator were proposed in [START_REF] Li | A compact UWB bandpass filter with improved upper-stopband performance[END_REF][START_REF] Wong | Ultra-wideband (UWB) microstrip bandpass filters with improved upper-stopband and miniaturized size[END_REF][START_REF] Wong | EBG-embedded multiple-mode resonator for UWB bandpass filter with improved upperstopband performance[END_REF][START_REF] Li | Compact UWB bandpass filter using stub-loaded multiple-mode resonator[END_REF] aiming to allocate the first resonant frequencies within 3.1-10.6 GHz band while suppressing spurious harmonics at the upper-stopband. In [START_REF] Kuo | High selectivity ultra-wideband (UWB) multimode stepped-impedance resonators (SIRs) bandpass filter with two-layer broadside-coupled structure[END_REF], quasi-Chebyshev bandpass filter with nine transmission poles is synthesized using three triple-mode SIRs two-layer broadside-coupled structure. Ground plane aperture techniques, increasing coupling between pairs MMR resonators, were also widely investigated to improve the out-band rejection in UWB filters [START_REF] Kuo | High selectivity ultra-wideband (UWB) multimode stepped-impedance resonators (SIRs) bandpass filter with two-layer broadside-coupled structure[END_REF][START_REF] Wang | Aperture-backed microstrip line multiple-mode resonator for design of a novel UWB bandpass filter[END_REF]. Furthermore, stub bandpass filters with several configurations could be also used to meet UWB mask requirements [START_REF] Wong | Highly selective microstrip bandpass filters for ultrawideband (UWB) applications[END_REF][START_REF] Shamman | A novel ultra-wideband (UWB) bandpass filter (BPF) with pairs of transmission zeroes[END_REF][START_REF] Cai | A compact UWB bandpass filter using two-section open-circuited stubs to realize transmission zeros[END_REF][START_REF] Gong | An improved design method for UWB filter using two-section open-circuited stubs[END_REF]. In [START_REF] Wong | Highly selective microstrip bandpass filters for ultrawideband (UWB) applications[END_REF][START_REF] Shamman | A novel ultra-wideband (UWB) bandpass filter (BPF) with pairs of transmission zeroes[END_REF] connecting lines between short-circuited stubs were meandered in order to reduce the filter size. These UWB filters have sharp rejection, but their spurious response would degrade the out-of-band response and an increase in number of sections may lead to large insertion loss, as well as poor group delay. Besides, stepped impedance open stubs were implemented on this kind of structures to realize transmission zeros and get sharper attenuations [START_REF] Cai | A compact UWB bandpass filter using two-section open-circuited stubs to realize transmission zeros[END_REF][START_REF] Gong | An improved design method for UWB filter using two-section open-circuited stubs[END_REF].

In addition, tight coupling in a microstrip to CPW or slotline transition has also been investigated [START_REF] Li | An ultra-wideband bandpass filter using broadside-coupled microstrip-coplanar waveguide structure[END_REF][START_REF] Wang | Ultra-wideband bandpass filters with hybrid microstrip/CPW structure[END_REF][START_REF] Kuo | Compact ultra-wideband filters using composite microstrip-coplanarwaveguide structure[END_REF][START_REF] Li | Ultra-wideband (UWB) bandpass filters with hybrid microstrip/slotline structures[END_REF][START_REF] Li | Compact UWB bandpass filter on hybrid microstrip/slotline structure with improved out-of-band performances[END_REF]. The use of microstrip-to-CPW transition has received great attention providing a wide bandpass operation with a good trade-off in terms of insertion loss and group delay flatness [START_REF] Li | An ultra-wideband bandpass filter using broadside-coupled microstrip-coplanar waveguide structure[END_REF]. In [START_REF] Wang | Ultra-wideband bandpass filters with hybrid microstrip/CPW structure[END_REF], the UWB filter with a multiple-mode resonator was proposed using the microstrip-to-CPW transitions as inverter circuits. Composite transition lines exhibiting cross-coupling and slow-wave behavior were later proposed in [START_REF] Kuo | Compact ultra-wideband filters using composite microstrip-coplanarwaveguide structure[END_REF], leading to more compact devices.

Recently, back-to-back microstrip-slotline cross-junction transition is rearranged to make up a novel class of UWB bandpass filter with lenient tolerance in fabrication [START_REF] Li | Ultra-wideband (UWB) bandpass filters with hybrid microstrip/slotline structures[END_REF][START_REF] Li | Compact UWB bandpass filter on hybrid microstrip/slotline structure with improved out-of-band performances[END_REF].

Basically, a wide bandwidth filter may be implemented by a direct cascade of the low-and high-pass filters. In [START_REF] Lin | Wideband coplanar-waveguide bandpass filters with good stopband rejection[END_REF], the wideband coplanar-waveguide (CPW) bandpass filters based on the cascade of CPW low-and high-pass periodic structures were constructed. To save the circuit area, a high-pass filter realized by shunt quarter-wave short-circuited stubs is embedded with a stepped-impedance low-pass filter [START_REF] Hsu | Microstrip bandpass filter for ultra-wideband (UWB) wireless communications[END_REF]. As a result of the use of low degree high-pass filter, the filter lacked selectivity or sharpness at the lower frequency. Note that a suspended stripline technology combining low loss with reduced size was also used to fabricate this composite filter [START_REF] Menzel | Low-loss ultra-wideband (UWB) filters using suspended stripline[END_REF]. Analytical techniques to synthesize an isolated cascade of high-and low-pass sections based on an iterative algorithm for both Butterworth and Chebychev cases were also reported in [START_REF] Menzel | Low-loss ultra-wideband (UWB) filters using suspended stripline[END_REF]. In our previous work [kaddour, EUMA], several UWB filters were realized in a semi-lumped approach using both surface mounted (SMT) capacitors and microstrip transmission lines. Thanks to the slow-wave behavior introduced by capacitively loaded transmission lines, a high degree of miniaturization is demonstrated in [START_REF] Kaddour | A compact and selective low-pass filter with reduced spurious responses, based on CPW tapered periodic structures[END_REF].

In this paper, a technique of spurious suppression is studied. The paper is organized as follows. First, a design method is carried out and validated by simulations in section II. Then, several planar filters with 3-dB fractional bandwidth of 142 % centered at 0.77 are designed in section III. Measurement and simulation results are in very good agreement. The insertion loss is limited to less than 1 dB in the passband, and the maximum group delay variation is 1.6 ns. Furthermore, a spurious suppression technique using capacitively loaded stubs is introduced.

Consequently, the first spurious frequency appears at more than eight times the center frequency when a 40-dB out-of-band attenuation is considered. Finally, twisted capacitively loaded stubs are used to minimize the circuit size (55%) providing a good agreement between measurements and simulations.

II. Principle A) Configuration and synthesis technique

Figure 1 shows the equivalent electrical circuit of the semi-lumped proposed UWB bandpass filter. Following the same principle as in [START_REF] Hsu | Microstrip bandpass filter for ultra-wideband (UWB) wireless communications[END_REF], a low-pass and a high-pass filters are combined. The high-pass filter is realized by shunt quarter-wave short-circuited stubs. The stepped impedance low-pass filter used in [START_REF] Hsu | Microstrip bandpass filter for ultra-wideband (UWB) wireless communications[END_REF] is replaced in this work by a low-pass filter based on capacitively loaded transmission lines as published in [START_REF] Kaddour | A compact and selective low-pass filter with reduced spurious responses, based on CPW tapered periodic structures[END_REF]. Comparing with the distributed filter [START_REF] Hsu | Microstrip bandpass filter for ultra-wideband (UWB) wireless communications[END_REF], a size reduction reaching 85% was demonstrated thanks to the use of the semi-lumped technology [Kaddour, EUMA]. In addition, two series capacitors are added at the near-and far-ends of the filter in order to give more sharp rejection around the lower stopband. In order to investigate the synthesis procedure, an UWB filter centered at 0.77 GHz with a 142% relative 3-dB bandwidth, is considered. In this paper, the center frequency f c of the bandpass filter is given by:

Z 0 Z 0 C p1 Z 0 Z 0 C s Z 0 Z 0 Z 0 C p2 C p2 C p1 C s θ 1 θ 1 θ 2 θ 2 θ 3 θ 4 θ 3 Z s θ s Z s θ s
H L c f f f = (1)
where f L and f H are respectively the low and high 3-dB cut-off frequencies.

The proposed synthesis procedure concerns the establishment of a rough initial estimation of the filter's parameters, by setting both low and high cut-off frequencies. To simplify the study, an initial design with ideal transmission lines and capacitors is adopted. In addition, identical series transmission lines' electrical lengths are first considered, i.e. θ= θ 1 = θ 2 = θ 3 = θ 4 . The main advantage of this filter topology is the independent control of both low and high cut-off frequencies. The upper and lower cut-off frequencies are respectively controlled by the low-pass and high-pass sections.

The low-pass sections are realized by capacitively loaded transmission lines following the same principle of the low-pass filters published in [START_REF] Kaddour | A compact and selective low-pass filter with reduced spurious responses, based on CPW tapered periodic structures[END_REF]. In this previous work, the low-pass section has been studied and the design equations relating the characteristic impedance Z 0 , the loading capacitance C p , the transmission line electrical length θ to the cut-off frequency f H were derived. For a characteristic impedance equal to 120 Ω and assuming a 1.5 GHz high cut-off frequency, the equations established in [START_REF] Kaddour | A compact and selective low-pass filter with reduced spurious responses, based on CPW tapered periodic structures[END_REF] lead to C p = 3.1 pF and θ = 29°@ 1.5 GHz.

While the high cut-off frequency f H is adjusted by the low-pass section, the low cut-off frequency f L is directly controlled by the short circuited stubs electrical length θ s . For thus, a simple tuning carried out with Agilent ADS [30] revealed that the electrical length θ s should be set to 23° @ 0.4 GHz.

Currently, rough initial values for C p , θ and θ s are available. The filter response simulated using the following parameters (C p = 3.1 pF, θ=29° @1.5 GHz, θ s =23° @0.4 GHz) is illustrated in Figure 2. As a result, the obtained cut-off frequencies are 0.4 GHz and 1.45 GHz, respectively. So the simple design method described above is sufficient for a rough estimation of the filter's parameters. Note that the simulated filter suffers also from a poor matching in the passband with a return loss reaching only 8 dB around 1.3 GHz. However, the return loss will be improved in the second step of the design when the filter will be optimized. For the second step consisting of a whole optimization of the complete filter, a microstrip technology is considered. A Rogers™ RO4003 substrate with relative permittivity ε r =3.38, substrate thickness h= 813 μm, and dielec-tric losses tanδ= 0.0027, has been adopted. Also, T junctions, surface mounted capacitor's, soldering pads and via holes models are taken in account. Complete models taking into account the parasitic series elements with an inductance of 0.35 nH and a resistance of 0.25 Ω are considered to get more accurate simulation responses. In order to maximize freedom degrees and improve filter performances, the series transmission lines' electrical length and the capacitors' value are no more considered as equal.

The UWB filter is optimized using Agilent ADS [30]. The transmission lines characteristic impedance has been set to 120 Ω, leading to a strip width W hi equal to 250 µm. The value for the series capacitors added in the nearand far-end of the UWB bandpass filter to improve the low stop-band rejection is fixed to 6.8 pF, leading to a small impedance compared to 50 Ω, keeping unchanged the low cut-off frequency. Table 1 lists the electrical lengths and capacitor's values obtained after optimization for the UWB filter. Figure 3 gives the simulation results to 8 GHz. As illustrated, return loss is better than 20 dB in the whole passband, and insertion loss is limited to 0.2 dB at the center frequency. The shape factor, defined between -3 dB and -20 dB, equals 1.3:1. In the simulated response, spurious frequency occurs around 5 GHz, leading to an out-of-band rejection level of -40 dB to only 4.5 GHz, i.e. 4.5 times the center frequency. The study of the spurious origin is carried out in the next section where a simple original technique to reject theses spurious frequencies is suggested. Frequency (GHz) 1.

Figure 3. Simulation response of the bandpass filters using parameters' values given in Table

B) Spurious analysis and improvement

The basic principle of spurious suppression is based on the transmission zeros repartition in the frequency response. As illustrated in Figure 4, three transmission zeros are observed respectively at 2.2, 3.2, and 6 GHz. So a possible solution to reject the spurious frequencies and improve the rejection band width could be to move the transmission zeros, either to push the 3.2 GHz transmission zero to higher frequencies, or to pull the 6 GHz transmission zero to lower frequencies. Therefore, the study of the origin of these transmission zeros has to be carried out first.

Transmission zeros located at 2.2 GHz and 3.2 GHz are due to the resonance of the LC shunt circuit formed by the SMT capacitor and the parasitic inductances. This parasitic inductance is the sum of the intrinsic SMT capacitor's, the soldering pads, and the via holes inductances. Considering the capacitor's model and the via hole's dimensions, its value L s can be estimated to 1.3 nH. The transmission zero location is then easily calculated using the expression of the LC resonant frequency:

pi s r C L f π 2 1 = . ( 2
)
Indeed, the 2.2-GHz transmission zero is introduced by the resonant circuit formed by the 3.9-pF capacitor and the series parasitic inductance L s =1.3 nH, while the 3.2-GHz transmission zero is related to 1.8-pF capacitor. As the high cut-off frequency is controlled by the LC circuit, the lower transmission zeros could not be moved. A modification of these transmissions zeros will affect the passband response of the UWB filter. The transmission zero appearing at 6 GHz is due to the short-circuited stub resonance that presents a 180° electrical length near to 6 GHz. A solution for spurious suppression would be to pull this transmission zero to lower frequencies. This could be easily achieved by increasing the stub electrical length. However, as the low cut-off frequency is fixed by the stub's length, the low frequency electrical length should not be modified. Thus, capacitively loaded stubs could be successfully used for spurious suppression, leading to the whole UWB bandpass filter equivalent electrical circuit given in Figure 5. A capacitively loaded stub will behave as an unloaded stub at low frequency because the capacitor's susceptance is high, so that the stub's behavior remains unchanged. When the frequency will increase, the loading capacitor becomes significant compared to the stub's transmission line distributed capacitor, and the wave velocity in the loaded stub will decrease, leading to a lower resonant frequency for the stub. Figure 7 compares the UWB filter insertion loss with both loaded and unloaded stubs. Whereas the filter response remains unchanged in the passband, the loading capacitor effect appears for higher frequencies. When the capacitively loaded stubs are used, the third transmission zero which was located at 6 GHz occurs now at 3.8 GHz, while the first two transmission zeros due to LC resonance remain fixed. Consequently, the high frequency rejection band, defined below an attenuation of -40 dB, is enlarged to about 6.5 GHz, i.e. more than eight times the filter's center frequency of 0.77 GHz. First, measurement and simulation results of the UWB filter with unloaded stubs are given in Figure 9. Comparison between the simulation and measurement results of the UWB filter with unloaded stubs. A good agreement is observed in the filter passband. A wideband of 1.1 GHz is achieved leading to the expected bandwidth of 142%.

Z 0 Z 0 Z 0 Z 0 C p1 Z 0 Z 0 Z 0 C p2 C p2 C p1 C s C s θ 1 θ 1 θ 2 θ 2 θ 3 θ 4 θ 3 θ s1
Z s θ s1 Z s C sp θ s2 Z s C sp
The measured return and insertion loss are found to be higher than 18 dB and lower than 0.3 dB over the passband, respectively. The first spurious response appears around 4.5 GHz. A slight shift of about 400 MHz is observed between simulations and measurements. This shift is probably due to the capacitor's model that is not sufficiently accurate at high frequencies. The measured rejection band, if a minimum 40-dB attenuation is considered, extends to about 4.2 GHz, i.e. 5.4 times the center frequency. Measurements of both filters with unloaded and loaded with SMT capacitors stubs are compared in Figure 10.

While the bandwidth is quite the same for the two topologies, return loss is improved to 20 dB with the capacitively loaded stubs. With the introduction of SMT capacitors, the first spurious peak is shifted from about Next, Figure 11 gives the comparison between the measurement results to 8 GHz, for both filters with stubs loaded by SMT and patch capacitors. With patch capacitors, the return loss is better than 18 dB in the passband and the rejection band is slightly enlarged from 6 to 6.6 GHz. So it can be concluded that capacitively loaded stubs can be successfully used for spurious suppression. For miniaturization purposes, bended stubs could be used. This technique could not be easily achieved with stubs loaded by patch capacitors. So, a filter with bended stubs loaded by SMT capacitors was fabricated. Figure 12 gives the photograph of the miniaturized filter. The miniaturization improvement with bended stubs reaches 55%

(in term of surface) compared to the filter with straight stubs. A good agreement between measurement and simulation results is observed in Figure 12. Return loss is better than 22 dB and the rejection band extends to about 6 GHz.

V. Conclusion

In this paper, a new topology of semi-lumped UWB bandpass filter has been proposed. It is based on the combination of a low-pass and a high-pass filters. The synthesis method of the bandpass filter has been developed and validated by simulations. Filters centered at 0.77 GHz with a relative bandwidth of 142% have been designed and fabricated. Simulated and measured results have shown a good agreement in terms of return loss (better than 18 dB) and insertion loss, (lower than 0.3 dB). A spurious suppression technique, using capacitively loaded stubs, has been carried out and has demonstrated its efficiency to eliminate spurious responses until more than eight times the center frequency. A miniaturization better than 55% has also been achieved by a simple stub bending.

Figure 1 .

 1 Figure 1. Equivalent electrical circuit of the proposed semi-lumped UWB filter.

Figure 2 .

 2 Figure 2. Simulation response of the UWB bandpass filter with Cp= 3.1 pF, θ H =29° @ 1.5 GHz, and θs =23°@ 0.4 GHz.

Figure 4 .

 4 Figure 4. Transmission zeros of the UWB bandpass filter till 8 GHz.

Figure 5 .Figure 6 .

 56 Figure 5. UWB filter equivalent electrical circuit with capacitively loaded stubs.

Figure 7 .Figure 8 .

 78 Figure 7. Simulated insertion loss for the UWB filter with loaded-and unloaded stubs.

Figure 9 .

 9 Figure 9. Comparison between the simulation and measurement results of the UWB filter with unloaded stubs.

4. 5 Figure 10 .

 510 Figure 10. Comparison between measured frequency responses, for both UWB filters with unloaded and loaded by SMT capacitors stubs.

Figure 11 .

 11 Figure 11. Comparison between measured frequency responses, for both UWB filters with capacitively loaded stubs.

Figure 12 .

 12 Figure 12. Simulations and measurements response for the UWB filter with bended capacitively loaded stubs.

Table 1 . Electrical lengths and capacitors' values of the proposed UWB semi-lumped bandpass filters.

 1

	θ 1 (°)	θ 2 (°)	θ 3 (°)	θ 4 (°)		θ s (°)	C p1 (pF)	C p2 (pF)	C s (pF)
	7		17	6	26		12	1.8		3.9	6.8
		S-parameters (S 21 , S 11 in dB)	-40 -30 -20 -10 0				|S 11 |S 21 | |
			0	1	2	3	4	5	6	7	8


Hui Zhang

Huazhong Shu

Gouenou Coatrieux

Jie Zhu

Jonathan Q M Wu

Yue Zhang

Hongqing Zhu

Limin Luo

Affine Legendre moment invariants for

Keywords: Affine transformation, geometric attacks, image watermarking, Legendre moment invariants

Introduction

Image watermarking has been proposed to respond copyright protection concerns , . To be efficient, a watermarking scheme [START_REF] Petitcolas | Watermarking schemes evaluation[END_REF] [START_REF] Podilchuk | Digital watermarking: Algorithms and applications[END_REF] must be robust against a wide variety of attacks. Among these attacks, geometric distortions are more difficult to tackle as they affect synchronization between the watermark reader and the embedder.

A number of algorithms robust to rotation, scaling, translation (RST) have been reported in the literature . Ruanaidh

[3 ]- [START_REF] Zheng | A survey of RST invariant image watermarking algorithm[END_REF] et al. [START_REF] Ruanaidh | Rotation, scale, and translation invariant spread spectrum digital image watermarking[END_REF] utilize the Fourier Mellin transform so that the watermark signal is not impacted by geometric distortions. Image normalization has also been proposed for watermark embedding/extraction in . In particular, Kim watermark Zernike moments of the normalized [START_REF] Tang | A feature-based robust digital image watermarking scheme[END_REF]- [START_REF] Kim | Invariant image watermark using Zernike moments[END_REF] et al. [START_REF] Kim | Invariant image watermark using Zernike moments[END_REF] image. Normalization allows scale and translation invariance while Zernike moments give robustness to rotation. But, as stated by the authors, it seems not possible to watermark directly Zernike moments. They adopt an iterative procedure to construct the watermark from the Zernike moments in the spatial domain in order to control watermark invisibility while imposing specific values to Zernike moments for watermark detection. The resulting watermark is then added to the image pixels. This scheme is public as the original image is not required for detection and has one bit capacity (see for a recent survey). It should be noted that image normalization may increase the [START_REF] Zheng | A survey of RST invariant image watermarking algorithm[END_REF] computation time and also induce errors in watermark detection/extraction due to image interpolation.

As a general case of RST transformation, affine transformation is more complex. In , a template constituted of local peaks at [START_REF] Pereira | Robust template matching for affine resistant image watermarks[END_REF] predefined position is embedded in the discrete Fourier transformed image for the purpose of detecting the affine transformation the watermarked image undergone. An invariant watermark proposed by Alghoniemy is based on affine geometric moment et al. [START_REF] Alghoniemy | Geometric invariance in image watermarking[END_REF] invariants , . They modify moment values of the image so that a predefined function of its geometric moment invariants, a [11 ] [12 ] weighted combination of them, lies within a predetermined value. This method is one bit watermarking and public. But, as for , a [START_REF] Kim | Invariant image watermark using Zernike moments[END_REF] memory and time consuming exhaustive search is necessary to adapt the strength of the watermark, added in the spatial domain, while preserving the output of the predefined function. In fact, moments and moment invariants used in above approaches cannot be watermarked directly. Dong exploited geometric moments and the corresponding central moments within an image et al. [START_REF] Dong | Digital watermarking robust to geometric distortions[END_REF] normalization procedure. The image is normalized so that it meets a set of predefined moment s criteria. The normalized image is ' consequently invariant to affine geometric transform. This latter is spread spectrum watermarked before being denormalized. This scheme is public and allows multi-bit watermarking but, as above, it may suffer of errors due to image interpolation.

Most of these methods make use of geometric moments which are not orthogonal. However, orthogonal moments are better in terms of image description and are more robust to noise . Consequently, it can be expected that a set of affine invariants derived from [START_REF] Teague | Image analysis via the general theory of moments[END_REF]- [START_REF] Shu | Moment-based approaches in image part 2: invariance[END_REF] orthogonal moments will offer better performance in terms of robustness, and allows direct watermarking of invariants avoiding thus iterative embedding. Although the orthogonal moments including pseudo-Zernike moments, Tchebichef moments and Krawtchouk moments have been already used to image watermarking , none of them takes the affine transformation into consideration.

[18 ]- [START_REF] Venkataramana | Image watermarking using Krawtchouk moments[END_REF] In this paper, we present a new method robust to geometric distortion. It is based on a set of orthogonal Legendre moment invariants we propose. The rest of this paper is organized as follows. Section II reviews the definition of Legendre moments and presents our set of invariants to image affine transformation. Watermark embedding, detection, and extraction processes are given in Section III. Before concluding, experimental results are provided in Section IV. They illustrate the overall performance of our approach.

Affine Legendre Moment Invariants

Legendre Moments Definition

The 2-D ( ) th-order Legendre moment of an image function ( ) is defined as p + q f x, y [START_REF] Teh | On image analysis by the method of moments[END_REF] where ( ) is the th-order orthonormal Legendre polynomial given by P p x p with It can be deduced from ( ) that 2 where ( ), 0 , is the inverse matrix of the lower triangular matrix ( ). The elements of are given by

D M = d p, k ≤ k ≤ p ≤ M C M = c p, k D M [21 ]
Using the orthogonality property of Legendre moments, the image can be approximately reconstructed from a finite number moments of order up to ( ) as M, M

Legendre Moments of an Affine Transformed Image

In this subsection, we establish the relationship between the Legendre moments of an affine transformed image and those of the original image. The affine transformation can be represented by [START_REF] Rothe | The method of normalization to determine invariants[END_REF] where is called the homogeneous affine transformation matrix.

Translation invariance can be achieved by locating the origin of the coordinate system to the center of mass of the object such that . Thus, ( , ) can be ignored and only the matrix is taken into consideration in the remaining part of this paper. However, this

x 0 y 0 simplification is not valid when considering image cropping attack as the center of mass will change (see Section IV).

The 2-D ( ) th-order Legendre moment of the affine transformed image g( , ) is defined by p + q x ′ y ′ where det ( ) denotes the determinant of the matrix .

A A

We can now link the Legendre moments of the affine transformed image given by ( ) with those of the original image. By replacing 8 the variable by in ( ), we have x a 11 x + a 12 y 2 Similarly Substituting ( ) and ( ) into ( ) yields 9 10 8

Using ( ), we have 4

Substitution of ( ) into ( ) leads to 12 11 shows that one Legendre moment of the transformed image is a linear combination of those of the original image. Equation ( 13)

Affine Legendre Moment Invariants (ALMIs)

Using ( ), we can derive a set of ALMIs but its direct use leads to a complex nonlinear system of equations. To reduce complexity, 13

we decompose the matrix into a product of simple matrices. Two kinds of decomposition known as XSR and XYS decompositions can be used , . In this work, we adopt the XYS decomposition, which consists in decomposing the affine matrix into an -shearing, a Based on this decomposition and using ( ), we derive through the following theorems a first set of Legendre moment invariants 13 and that are invariant to -shearing, -shearing and anisotropic scaling, respectively.

x y

Theorem 1

Let be an original image and its -shearing transformed version such as ( ) ( , ). Then the following are g x g x, y = f x + β 0 y y invariant to -shearing x

where is a parameter associated with the image such that The proof of Theorem 1 is given in the .

β f f β f = β g + β 0 appendix Theorem 2
Let be an original image and its -shearing transformed version such as ( ) ( ). Then the following are f g y g x, y = f x, γ 0 x + y invariant to -shearing: y

where is a parameter associated with the image such that Theorem proof is similar to that of Theorem 1 and is omitted

γ f f γ f = γ g + γ 0
here.

Theorem 3

Let and be two images having the same shape but distinct scale, i.e., ( ) ( , ). Then the following are invariant f g g x, y = f α 0 x δ 0 y to anisotropic scaling where and are two parameters associated with the image such that and . Theorem proof is given in the

α f δ f α f = α 0 α g δ f = δ 0 δ g . appendix
Determination of the parameters , , and will be discussed in Section II-D.

β f γ f α f δ f
Notice that we can also derive the following theorem without proof.

Theorem 4

The Legendre moments of an image can be expressed through a linear combination of their invariants as follows:

As we will show in Section III, this last theorem will be of great interest for watermarking as it allows avoiding iterative embedding.

From that standpoint, by combining that are, respectively, invariant to -shearing, -shearing, and anisotropic scaling, x y we can obtain our set of ALMIs. For an image ( ), we use the following process. f x, y

Step 1 -shearing Legendre moment invariants are calculated by ( ), where the Legendre moments are computed with ( ). x 15 1

Step 2

The combined invariants with respect to -shearing and -shearing are computed by ( ) where the Legendre moments on the x y 16

right-hand side of ( ) are replaced by computed in Step 1. 16

Step 3

The affine Legendre moment invariants are calculated by ( ) where the Legendre moments on the right-hand side of ( ) are 17 17 replaced by computed in Step 2.

Parameter Estimation

As described above, the parameters , , , and in ( ) ( ) are image dependant. We provide one way for estimating these

β f γ f α f δ f 15 -17
parameters. Considering an affine transform and its XYS decomposition, by setting in ( ), we have 15

The parameter can then be determined by solving ( ). The parameters , , , and associated with the transformed image ( ) can also be estimated according to ( ), ( ), and (

β g γ g α g δ g g x, y 21 23
). It can be verified that the parameters provided by the above method satisfy the following relationships: , , 24

β f = β g + β 0 γ f = γ g + γ 0 α f =

and

, where , , , and are the coefficients of the affine transform applied to . Based on these relationships,

α 0 α g δ f = δ 0 δ g α 0 δ 0 γ 0 β 0 f
conditions given in theorems 1 to 3 are satisfied. It is worth noting that other choice of parameters can also be made to keep the invariance of ( ) ( ) to image transformation. For a detailed discussion on the parameter selection methods, we refer the readers to and . 15 -17 [START_REF] Zhang | On the choice of consistent canonical form during moment normalization[END_REF] [24 ]

Implementation Strategies

In this section, we describe the different processes for watermark embedding, detection and extraction. Proposed ALMIs can be watermarked directly and in different ways by applying spread spectrum or quantization index modulations. However, in order to conduct a fair comparison with other methods based on image moments, we decided to follow the procedure proposed by Alghoniemy . et al [START_REF] Alghoniemy | Geometric invariance in image watermarking[END_REF] for watermark detection. Our embedding procedure differs from their proposal as we can directly watermark image invariants contrarily to where an iterative procedure is adopted.

[10 ]

Watermark Embedding

Herein, watermark embedding is carried out in the Legendre moment invariants directly. To illustrate this, let us take the anisotropic scaling invariants as an example. The -shearing, -shearing and affine Legendre moment invariants and can be x y treated in a similar way.

As in , the watermark is generated from the Legendre moment invariants before being inserted in the invariant domain of the [START_REF] Alghoniemy | Geometric invariance in image watermarking[END_REF] original image. Watermark embedding can be noted as follows where and denote respectively the anisotropic scaling Legendre moment invariants of the original image and of its f watermarked version , and are the parameters of strength which are selected to achieve the best tradeoff between robustness and h s pq imperceptibility. In general, they are selected in a way such that the peak signal-to-noise ratio (PSNR) between the original image and the f watermarked image is larger than 40 dB in order to make the watermark invisible. The PSNR between and is defined as

h f h
where is the image size. N × N In this paper, a simple choice of consists to take for any and . It should be noticed that the watermark embedding method s pq s pq = s p q proposed by Kim corresponds to a special case of our method with and 0 for other value of and . et al. [START_REF] Kim | Invariant image watermark using Zernike moments[END_REF] s 31 = s 42 = s s pq = p q illustrates the watermarking of the reference image Lena see using its 210 first image moments and 0.0214 for Fig. 1 We can express the watermarked image as a function of the Legendre moment invariants of the original image. In fact, using ( ), (

) can be rewritten as 26

With the help of ( ), we have 6

where is the maximum moment or moment invariant order used for watermarked embedding, is the image associated with the M W watermark. The relationship between , and the PSNR is illustrated in . It can be seen that the PSNR obviously decreases with s Fig. 3 increasing values of while the order of moment invariants exploited for embedding has a little effect on the PSNR. s M

In this experiment, the embedded watermark is completely dependent on the image without any random component; it can be easily estimated from the watermarked image and removed. Thus the embedded watermark does not provide any security. However, this is only a limitation of this experiment as the watermark can be defined more secretly. In fact, instead of deriving the watermark from the Legendre moment invariants second term in ( ) , one can use a secret watermark pattern. We illustrate that capability in Section III-C, by adding a logo to obtain the watermarked image ( ).

B h = f +W = f + sB

Watermark Detection

Watermark detection aims at determining if the received test image is watermarked or not in order to prove ownership. Herein, we follow the same strategy as Alghoniemy . We use the distance between the two sets of moment invariants, i.e., between the et al. [START_REF] Alghoniemy | Geometric invariance in image watermarking[END_REF] ALMIs of the watermarked image and those of the received image as detector. The distance between two images in the feature space is measured by where ( ) and ( ) correspond to the result of a function applied to the ALMIs of the received and watermarked images, I t I h I respectively. As in , the function we retain is the mean function [START_REF] Alghoniemy | Geometric invariance in image watermarking[END_REF] I where is the total number of affine invariants used for detection. L

The detection decision is then made by comparing the distance ( ) with a predefined threshold . If the value of ( ) is d t, h d th d t, h smaller than , the detected watermark is declared authentic; otherwise, it is declared unauthentic. As defined, the original image is not d th required for the watermark detection but as this later relies on ( ) (i.e., a digest of the watermarked image), this method is one bit I h

watermarking and semi-blind.

Watermark

The procedure we follow in order to recover the watermark from a received image is given in . For simulating this, we consider Fig. 4 that a watermarked image has been affine attacked becoming a received image . To sum up, once the watermark detected in , we h t t estimate the affine transform coefficients , , , and . A restored image can be derived from by inverting the estimated

α 0 δ 0 γ 0 β 0 h ′ t
transform. One has just to subtract the original image from to get access to the watermark . Consequently and contrarily to the

f h ′ W ′
detection process, the watermark extraction procedure is private, as it requires the original image.

Coefficients , , , and of the affine transform can be estimated in the following way. Let denote ( ) and ( ) be the

α 0 δ 0 γ 0 β 0 M h M t
parameter matrix associated to and , respectively h t ( ) and ( ) parameters can be estimated through the procedure given in Section II-D, by making use of ( ) ( ). With these M h M t 21 -25 notations, , , , and are directly given by the product ( ) and ( ).

α 0 δ 0 γ 0 β 0 M h M 1 -t

Experimental Results

Eight standard gray images of 256 256 pixels shown in were used to evaluate the performances of our scheme. For these × Fig. 2 experiments, was set to 20 (i.e., 210 moments were used for embedding). For the comparison purpose, the same invariants order was M ' considered in our scheme and the one proposed by Alghoniemy for embedding. Furthermore, for both methods, moment invariants of et al.

order up to three were used for watermark detection i.e., 4 in ( ) . More clearly, , , , and were used since

β f γ f α f δ f [ -21 -25 ]
In a first experiment, was set to 0.0214, 0.0192, 0.0189, 0.0198, 0.0192, 0.0237, 0.0179, and 0.0187 for Lena, Cameraman, Woman, Boat, Gold Hill, Bridge, Harbor, and Girl images in order to achieve a PSNR of 40.00, 40.01, 40.02, 40.06, 40.00, 40.01, 40.02, and 40.01 dB, respectively. Parameters of were fixed in order to get equivalent PSNR values. Four types of distortions have been considered: [START_REF] Alghoniemy | Geometric invariance in image watermarking[END_REF] rotation, scaling, additive Gaussian noise and JPEG compression. For image rotation, we apply angles varying from 0 to 120 every 20. For image scaling, we consider scale factor evolving from 0.1 to 0.6 with a step of 0.1. The standard deviation of the Gaussian noise varies from 5 to 30 every 5 for the additive noise attack. The JPEG compression quality factor varies from 10 to 60 with a step of 10. We give the average variation of the distance ( ) used in the detection process, i.e., the distance between ALMIs of the watermarked image and of 30 h the received image , for the eight test images. It can be seen from that with respect to rotation, scaling, additive Gaussian t Fig. 5(a) (d)

noise and JPEG compression our ALMIs have a better behavior than the affine geometric moment invariants (AGMIs) adopted by Alghoniemy . ALMIs variability is also smaller than AGMIs. In fact, we achieved an averaged standard deviation of 0.2 for et al. [START_REF] Alghoniemy | Geometric invariance in image watermarking[END_REF] ' % ALMIs against 2.6 for Alghoniemy s method in all these experiments.

% '

Considering the same test image set, we then compare detection performance of our scheme with for different PSNR values and [START_REF] Alghoniemy | Geometric invariance in image watermarking[END_REF] under various attacks including rotation, scaling, affine transformation, median filtering, Gaussian noise, salt and pepper noise, speckle noise, small random distortions (SRD), JPEG compression, cropping, and histogram equalization. For that purpose, we used stirmark 4.0 I ratio between the number of correctly detected watermark and the number of tested image; and, the average detection distance for and [START_REF] Alghoniemy | Geometric invariance in image watermarking[END_REF] ALMIs see ( ) under the attacks described above. It can be seen that the proposed method achieves better results whatever the attack type. However, as our scheme is not robust to cropping and histogram equalization attacks. This may be explained by the fact that: 1) [START_REF] Alghoniemy | Geometric invariance in image watermarking[END_REF] we use the image center of mass as origin of the coordinate system, center of mass usually modified by such kind of modifications (see Section II-A), and 2) changes of the image intensity more or less impact invariants values.

'

As shown in Section III-C, once the watermarked is detected, one can estimate the affine transform parameters allowing then the watermark extraction. To illustrate the efficiency of our system in that situation, we use a logo image as watermark

B W: h = f + sB = f + W
, where , , and correspond to the watermarked, the original and the logo images, respectively (see Section III-A). is of same h f B B dimensions than our test images, see and was embedded in the four test images shown in using all image moments

Conclusion and Perspectives

The major contribution of this paper relies on two aspects. The first one is the derivation of a set of affine invariants based on Legendre moments. Those invariants can be used for estimating the affine transform coefficients applied to one image. The second one is the use of these affine Legendre moment invariants for watermark embedding, detection and extraction. It was shown that the proposed method is more robust than others based on geometric moments.

One weak point of this algorithm is that the watermark detection is considered as a 1-bit watermarking system since the distance between the affine invariants and the threshold is used. However, the proposed detection approach could be extended to a multi-bit watermarking scheme by making use of spread spectrum techniques for example. This subject is currently under investigation. Another limitation of the proposed algorithm is that it is not robust to image cropping and histogram equalization, a common problem for the moment-based watermark algorithms.

where Substitution of ( ) into ( ) yields A3 A2

Note that we have shifted the indices in the last step of ( ). Using ( ), we have A5 A4

Since both and are lower triangular matrices, and is the inverse of matrix , we have

C M D M D M C M
Substituting ( ) into ( ) leads to A7 A6

Using the relationship , we obtain

β f =β g + β 0
It can be deduced from ( ) and ( ) that . A9 A5

Proof of Theorem 3

From ( ), we have 13 can be written in matrix form as Equation [START_REF] Shu | Moment-based approaches in image part 2: invariance[END_REF] can also be written in matrix form as Equation (A10) Substituting ( ) into ( ) and using the relationships , and where is the th order identity matrix, we A12 A11 D C p p = I p α f = α 0 α g δ f = δ 0 δ g I p p obtain Thus, we have .

Footnotes:

1 Online . Available: ---

[ 22 ]

 22 [START_REF] Zhang | On the choice of consistent canonical form during moment normalization[END_REF] Ax y -shearing and an anisotropic scaling matrix, that is

Fig. 2 (

 2 Fig. 2(a) ] s = a PSNR of 40 dB. The difference of and , i.e., the watermark, is depicted in . Sample values of were Figs. 2(a) 1(a) Fig. 1(b) Fig. 1(b) multiplied by 50 to enhance the difference.

  the value we retained for , , , and , and see algorithm of Section II-D ( ) ( ) . 11 = I 30 = I 20 = I 02 =

[Fig

  Fig. 6(a) ] Fig. 2(a) (d) -( 255) and fixed to 0.005. Watermarked images were then attacked by an affine transformation as illustrated in . Affine M = s Fig. 6(b) (e) coefficients and their estimations based on ( ) ( ) are listed in . It can be seen that our method fits well the affine 21 -25 Table II transformation coefficients. Extracted watermarks are shown in . They are correctly recovered. Fig. 6(f) (i) -

Fig. 2

 2 Fig. 1 (a) Watermarked image with PSNR 40 dB. (b) Magnified watermark.=

Fig. 3 Fig. 4

 34 Fig.3PSNR variation for the reference image Lena with respect to the parameter of strength ( ) and the invariants order ( ) used for embedding.s M

Fig. 5

 5 Fig. 5 Variation in average of the distance between moment invariants with respect to our image test set and different image attacks: (a) rotation attack; (b) scaling attack; (c) gaussian noise attack; (d) JPEG compression attack.

Fig. 6 (

 6 Fig. 6 (a) Logo used as watermark. (b) (e) Watermarked images under affine transformation. (f) (i) Extracted watermark from (b) (e).

  1 and Matlab 7.1. The threshold used to decide whether or not an image is watermarked was set to 0.02. The parameters of the two affine respectively. In average on the test image set, was set to 0.0250, 0.0199, 0.0110, and 0.0088 in order to achieve a PNSR of 38, 40, 45, s respectively. Results achieved with both methods are summarized in . Indicated values correspond to the detection rate, i.e., the Table

	transform attacks given in	Table I	are	a 11 =	1.1,	a 12 =	0.2,	a 21 =	0.1,	a 22 =	0.8, and	a 11 = -	1.2,	a 12 =	0.3,	a 21 =	0.4,	a 22 = -	0.6,

[START_REF] Teague | Image analysis via the general theory of moments[END_REF]

[START_REF] Teague | Image analysis via the general theory of moments[END_REF]

α 0 δ 0 γ 0 β 0

[START_REF] Teague | Image analysis via the general theory of moments[END_REF]

[START_REF] Teague | Image analysis via the general theory of moments[END_REF]

[START_REF] Teague | Image analysis via the general theory of moments[END_REF]

[START_REF] Teague | Image analysis via the general theory of moments[END_REF]

[START_REF] Teague | Image analysis via the general theory of moments[END_REF]

[START_REF] Teague | Image analysis via the general theory of moments[END_REF]

[START_REF] Teague | Image analysis via the general theory of moments[END_REF]

[START_REF] Teague | Image analysis via the general theory of moments[END_REF]

Acknowledgements:

This work was supported by the National Basic Research Program of China under Grant 2011CB707904, the NSFC under Grant 61073138, Grant 60975004, and Grant 60911130370, the Natural Science Foundation of Jiangsu Province of China under Grant SBK200910055 and Grant BK2010426, and the Key Laboratory of Computer Network and Information Integration (Southeast University), Ministry of Education. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Min Wu.

Appendix Proof of Theorem 1

The Legendre moment invariants of the image intensity function (

) is defined as g x, y

Now we want to prove . To that end, we have