Low valent, ligand stabilized (phosphines, N-heterocyclic carbenes = NHC) Ni(0) complexes are important species, being catalytically relevant in many transformations, such as cross coupling processes. 1 Among the known Ni(0) complexes, stable [Ni(L)(alkene) 2 ] or [Ni(L)(bis-alkene)] species (L=phosphines, NHC) are rare because of a lack of efficient, versatile synthetic route or availability of the appropriate precursors.

2 Indeed, [Ni(phosphine)(C 2 H 4 ) 2 ] have been reported by displacement of the poly-ene ligand from the extremely sensitive [Ni(CDT)] precursor (CDT = 1,5,9-cyclododecatriene), whereas other [Ni(L)(bis-alkene)] species have been prepared from air, light and heat sensitive [Ni(COD) 2 ]. [3][4][5][6][7][8] In turn, both Ni(0) complexes [Ni(CDT)] and [Ni(COD) 2 ] require the use of pyrophoric alkylaluminium derivatives to be synthesized. It is thus desirable to devise a rational, experimentally simpler, versatile method to access families of Ni(0) species.

Recently, bis(trimethylsilyl)-1,4-tetramethyl-dihydropyrazine 1 has been shown to behave as a one or two-electron reducing agent for numerous transition metal complexes. 9,10,11 It was also used to reduce several late transition metal halide precursors to the corresponding metallic nanoparticles in the absence of stabilizing ligand. 12 In the case of Ni, nanoparticles active in carbon-carbon cross coupling process were obtained. We have hypothesized that if the in situ generated reduced Ni center could be efficiently trapped by ligands the aforementioned desirable Ni(0) complexes would be readily accessible. In the present work, we show that from the simple, commercially available, [Ni(DME)Cl 2 ] precursor (DME = dimethoxy-ethane), several Ni(0) complexes could be synthesized in one pot. We also report that with strongly donating bis-phosphine ligands, the reaction leads to Ni(I) complexes instead. Please do not adjust margins Please do not adjust margins trapped. The next level of complexity concerns the selective stabilization by two different types of ligands. Indeed, in order to be able to generate efficient catalytic species, it is crucial for the metal to bear strongly bound ligands as well as weakly bound ligands that will be displaced during the transformation. For example, Lappert et al. studied the coordination of 1,3-divinyltetramethyldisiloxane (dvtms) to Ni(0) centers in order to synthesize hydrosilylation catalysts. 6,8 For that purpose, they synthesized [Ni(PR 3 )(dvtms)] This observation raised further mechanistic questions. Mashima and coworkers have investigated the reduction process using species 1 with several metal centers. They have shown that despite the fact that 1 is a potential 2-electron reducing agent, an excess (2 equiv.) of 1 was needed to reduce efficiently Ni(II) into Ni(0) nanoparticles and only half of reductant 1 was converted. 12 They have moreover proposed that coordination of the reducing agent is needed for the electron transfer and TMSCl elimination. 9

It was therefore of interest to investigate the mechanism of the reduction of a Ni(II) complex, using both NMR and EPR spectroscopies. We chose to study the reduction of [Ni(DME)Cl 2 ] in presence of one equiv. of PCy 3 and two equiv. of dvtms into complex [Ni(PCy 3 )(dvtms)] 2a. Most interestingly, with one equiv. of species 1, complex 2a is formed with a ca 50% yield ( 31 P NMR) within 5 minutes at room temperature, along with ca 50% of free PCy 3 . Interestingly, at this point already, the 1 H NMR spectrum is very poorly resolved, which prevented quantification of reagent 1 consumption, and points to the formation of paramagnetic species. After 30 min the intensity of the signal had drastically decreased to a very poorly defined 31 P NMR spectrum (ESI). A well-defined EPR spectrum was observed, consistent with the formation of Ni(I) species. Indeed, the room temperature X-band EPR solution is triplet with hyperfine coupling to two equivalent phosphorus nuclei. Simulation of this spectrum gives a g value of 2.143 and a coupling constant with P of 73 G. This data is fully consistent with literature precedence for [(bidentate phosphine)NiCl] species. 15 Formation of [Ni(PCy 3 ) 2 Cl] either as a monomer or dimer is therefore evidenced during the process, following a comproportionation mechanism between generated Ni(0) and Ni(II) precursor. 16 Nonetheless, when a second equiv. of 1 was added, the quantitative formation of complex 2a was observed by Please do not adjust margins Please do not adjust margins

A major difference can be seen between the complexes featuring mono and bidentate ligands. Indeed, starting from [Ni(dcpp)Cl 2 ], a stable Ni(I) dimer is formed. The four-coordinate Ni center adopts a square planar geometry and it is likely that the bulkiness of the ligand prevents coordination of the reducing agent necessary to allow further electron transfer.

In summary we have demonstrated the potential of organic reducing agent 1 to synthesize stabilized Ni(0) complexes from Ni(II) halides. We have shown the versatility of this method toward diversified ligands such as alkenes, tertiary phosphines and NHCs. Mechanistically, a two electron reduction of the Ni(II) precursor to Ni(0) followed by comproportionation to Ni(I) species is involved. When a monodentate trialkyl phosphine ligand is used, further reduction to Ni(0) is possible, while with the strongly donating bidentate ligand dcpp, the reaction stops at the Ni(I) stage.

Preliminary experiments showed promising results in the use of this method to access active species in various catalytic transformations.

  The reduction of [Ni(DME)Cl 2 ] with 2 equiv. of bis(trimethylsilyl)-1,4-tetramethyldihydropyrazine in presence of a ligand L and an excess of olefin cleanly leads to [Ni(L)(alkene) 2 ] complexes. When reduction is done in presence of 1,5-cyclooctadiene (COD), [Ni(COD) 2 ] is obtained. Such approach also allows access to the Ni I dimer [Ni(bis(dicyclohexylphosphino)propane)Cl] 2 .

(

  Scheme 2 Synthesis of [Ni(PR 3 )(dvtms)] complexes (2a-2d) Complexes 2a-d were characterized by 1 H, 13 C and 31 P NMR in accordance with literature. 8 The structure of complex 2d was not known and is presented here (Figure 1). The local geometry at the three-coordinate nickel atom is planar and the angles P-Ni-L 1 and P-Ni-L 2 (with L 1 and L 2 the centers of the coordinated double bounds) of 116.9° and 113.7° confirm the slightly distorted trigonal structure. The six-membered ring (Ni-C13-Si-O-Si-C18) adopts a chair structure. 6 The CC bond lengths of the olefins ligands measured at 1.40 Å (average) attest here the back donation from the nickel center into the π* orbitals of the coordinated olefins.

Figure 1 Scheme 3

 13 Figure 1 ORTEP drawings of 2d, Hydrogen atoms were omitted for clarity. Bond distances [Å] and angles [°] : Ni-P 2.2284(10), Ni-C14 2.010(3), Ni-C13 2.018(3) Ni-C19 2.008(3), Ni-C18 2.030(3), C13-C14 1.402(4), C18-C19 1.404(4), P-Ni-C13 133.98(9) , P-Ni-C14 93.36(9), P-Ni-C19 96.585(10), P-Ni-C18 137.27(9).

Figure 2 Scheme 4 14 Scheme 5

 24145 Figure 2 ORTEP drawings of 3, hydrogen atoms were omitted for clarity. Bond distances [Å] and angles [°] : Ni-P 2.1873(10), Ni-C22 2.000(4), Ni-C21 1.987(4), Ni-C23 1.985(4), Ni-C24 1.988(4), C22-C21 1.392(6), C23-C24 1.389(6), P-Ni-C22 95.89(12), P-Ni-C21 136.69(13), P-Ni-C23 137.14(15), P-Ni-C24 96.26(13), C21-Ni-C23 86.01(19).Having shown on several examples the efficiency of the reduction method for the synthesis of phosphine stabilized Ni(0) complexes, we then turned our attention to the related NHC complexes. We chose the known complex 4 as a representative example. Complex 4 has been obtained by COD substitution in [Ni(COD) 2 ] by NHC IPr(1,3-bis(2,6-diisopropylphenyl)imidazol-2-ylidene) in the presence of styrene with a 90% yield.7 Our one pot synthesis from [Ni(DME)Cl 2 ] (Scheme 4) allowed the isolation of complex 4 in 44% yield.

31PScheme 6

 6 Scheme 6 Proposed reduction mechanism

We thank Lionel Rechignat for EPR Measurements and IFP Energies nouvelles for financial support.

Notes and references


Teddy Furon

Watermarking error exponents in the presence of noise: The case of the dual hypercone detector

Keywords:

The study of the error exponents of zero-bit watermarking is addressed in the article by Comesana, Merhav, and Barni, under the assumption that the detector relies solely on second order joint empirical statistics of the received signal and the watermark. This restriction leads to the well-known dual hypercone detector, whose score function is the absolute value of the normalized correlation. They derive the false negative error exponent and the optimum embedding rule. However, they only focus on high SNR regime, i.e. the noiseless scenario.

This paper extends this theoretical study to the noisy scenario. It introduces a new definition of watermarking robustness based on the false negative error exponent, derives this quantity for the dual hypercone detector, and shows that its performances is almost equal to Costa's lower bound.

INTRODUCTION

This paper is a theoretical study of the performances of a precise zero-bit watermarking scheme. It is theoretical because it considers an unrealistic model where the signals to be watermarked and the noise are assumed to be Gaussian distributed and infinitely long. It is specific to a given watermarking scheme as it focuses on the hypercone detector. This watermark detection scheme is important as [START_REF] Sabbag | Optimal Watermark Embedding and Detection Strategies Under Limited Detection Resources[END_REF] proves its optimality under some conditions.

Nevertheless, the performances of this scheme and its optimal watermark embedding are known only in the "high SNR regime" [START_REF] Comesana | Asymptotically Optimum Universal Watermark Embedding and Detection in the High-SNR Regime[END_REF], i.e. when the attack noise power tends to zero. The main contribution of this paper is a follow-up extending paper [START_REF] Comesana | Asymptotically Optimum Universal Watermark Embedding and Detection in the High-SNR Regime[END_REF] to any SNR regime. A shift of paradigm makes this extension tractable: Instead of optimising the performances of the scheme for a given noise

ì X ì Z Detector ì Y ì R Embedder H 0 H 1 d ∈ {0, 1}
Figure 1: Zero-bit watermarking. The embedder hides a mark into the content. The detector checks for the presence of this mark.

power, the goal is to keep the performances acceptable over a maximum range of noise power. This shift brings a new viewpoint of this problem. Its gives birth to a new definition of watermarking robustness, which is a second contribution of the paper. As a minor contribution, this paper revisits as well the noiseless setup (i.e. the limit of the high SNR regime). Failing detecting the watermark in the noiseless setup is equivalent to failing watermarking of a given host signal. The amount of watermark power is not big enough to make that host signal detectable. This observation eases the computation of the false negative error exponent thanks to a rolling-ball region filtering. It also has a nice connection with isoperimetric Gaussian inequality.

Section 2 introduces zero-bit watermarking and the theoretical setup. Section 3 lists the assumptions and the requirements specific to digital watermarking. The paper starts by revisiting in Sect. 4 the noiseless setup originally considered in [START_REF] Comesana | Asymptotically Optimum Universal Watermark Embedding and Detection in the High-SNR Regime[END_REF], and extends this piece of theory to the noisy setup in Sect. 5. Section 6 proposes some upper and lower bounds adapting the rationale of M. Costa [START_REF] Costa | Writing on dirty paper (Corresp.). Information Theory[END_REF] to zero-bit watermarking. At last, a practical embedding strategy is deduced from this theoretical study in Sect. 7.

THE THEORETICAL SETUP

Zero-bit watermarking is different from multi-bit watermarking. While people usually knows what watermarking means, some get confused between the detection and the decoding of a watermark. In multi-bit watermarking, a first algorithm, so-called embedder, hides a message (possibly encoded in several bits) into a piece of content. A second algorithm analyses a piece of content and proceeds to a decoding. The decoding outputs the hidden message or the decision that the piece of content under scrutiny is indeed not watermarked.

In zero-bit watermarking, one is solely interested in distinguishing watermarked from non watermarked content. Therefore, the embedding does not hide any message, but just a mark. There is no modulation of a signal by the message to be transmitted since there is no message. Hence, the term zero-bit watermarking. In the same way, the second algorithm does not perform a decoding, but a detection of the presence or the absence of the mark (see Fig. 1).

Notations

A feature vector in R n is extracted from a piece of multimedia content. Vectors ì

x and ì r denote respectively the extracted features from an original content, so-called the host, and from the content received by the detector. The embedder transforms ì

x into ì y by adding a watermark signal: ì y = ì x + ì w(ì x). This vector depends on the host (for a side-informed watermarking scheme) and on a secret key (not indicated to keep notations simple).

We consider a power constraint watermark problem where the energy of the watermark per sample is limited.

∥ ì w(ì x)∥ 2 ≤ nP, ∀ì x ∈ R n . (1)
The Euclidean norm of vector ì x ∈ R n is denoted by ∥ ì x ∥. The model of an attack is the addition of a noise vector ì z, and the received vector extracted from the content under scrutiny is ì r = ì y + ì z. At the detection side, two hypotheses are competing. The decision of the detector is denoted by d: d = 1 if the received content is deemed watermarked, d = 0 otherwise. There are two types of errors:

Under H 0 : The received vector has not been watermarked:

ì r = ì x + ì z.
A false positive happens when d = 1 with probability

P fp := P[d = 1|H 0 ].
Under H 1 : The received vector has been watermarked: ì r = ì x + ì w(ì x)+ì z. A false negative happens if d = 0 with probability

P fn := P[d = 0|H 1 ].
To take a decision, we assume that the detector first computes a score from received vector s(ì r ) with s(•) : R n → R. Then, it compares this score to a threshold τ : d = 1 if s(ì r ) ≥ τ and d = 0 otherwise. This defines the region W ⊂ R n of the vectors deemed as watermarked:

W := { ì x ∈ R n |s(ì x) ≥ τ }. (2)

Theoretical setup

The theoretical setup assumes that the signals are instances of a white Gaussian distribution in R n . Denote by ì X the random host vector whose power is σ 2 X , ì X ∼ N ( ì 0 n , σ 2 X .I n ), and ì Z the random noise vector of power

σ 2 Z , ì Z ∼ N ( ì 0 n , σ 2 Z .I n ).
We assume that ì Z is independent of ì X and the secret key. Computation of the performances (P fp , P fn ) is difficult even under this simple setup. To facilitate comparison, the study focuses on the error exponents, i.e. the exponential decay rate of the error probabilities:

E fp := lim n→∞ - 1 n log P fp , E fn := lim n→∞ - 1 n log P fn . (3)
For the sake of simplicity, notations omit the fact that (E fp , E fn ) depends on (P, σ 2 X , σ 2 Z ). Note that, at then end of the paper, Section 7 deals with a more practical setup where no statistical model of the host is assumed.

REQUIREMENTS AND PRIOR ART

The prior art of zero-bit watermarking is mainly organised around the issue of obliviousness. When describing a watermarking scheme, assumptions about what the embedder and the detector know and do not know about the setup is critical. This matters when turning a theoretical watermarking scheme into a practical technique watermarking content.

Assumption on obliviousness

Multimedia contents have a wide diversity. Features extracted from these contents are certainly not white Gaussian distributed with a fixed power σ 2 X . This is the reason why the above setup is pure theory. As a small step towards being more realistic, content diversity may imply that σ 2 X vary from one content to another. A watermarking scheme relying on the knowledge of the watermark detector about that parameter (to guarantee a given P fp , for instance) is not applicable in practice.

This reasoning holds as well for P. The watermarking power usually depends on the masking properties of the host content. These properties are also very diverse from one content to another. For instance, the human eye is less sensitive to noise in textured areas than in flat regions. Therefore, one has to adapt the watermark power to the visual content of an image. The masking properties of the host content are usually analysed by a Human Visual System model at embedding. Yet, the watermark detector might receive a heavily distorted copy of the content preventing such analysis. In other words, the detector cannot know the value of P used at the embedding.

As for σ 2 Z , obliviousness is also a plus at the embedding side: The embedder may not know in advance the amount of noise power that the watermarked content will support.

To conclude, this paper integrates the specificities of watermarking in the theoretical setup by imposing the following obliviousness assumptions:

• The embedder is oblivious w.r.t. σ 2 Z . • The detector is oblivious w.r.t (σ 2 X , σ 2 Z , P).

Requirements

This section outlines the relevance of the concept of error exponent in practice where the length n is large enough. C. E. Shannon indeed motivated its use in his seminal work [START_REF] Shannon | Probability of error for optimal codes in a Gaussian channel[END_REF]. He warns that an error exponent a priori leads to inaccurate probability estimate:

As P fp = e -nE fp +o(n) , neglecting the term o(n) may cause large multiplier uncertainty. Yet, for given E fp and P fp , Shannon outlines thatlog(P fp )/E fp sharply determines the necessary vector length. There is thus a trade-off between the exponent E fp and the complexity of the scheme reflected by length n.

In practice, the main requirement is the probability of false positive P fp . In many applications, its level is low and must be provably low. This means that one has to prove that the detector operates at a required low level. In the theoretical setup, operating a given exponent E fp = E then determines the vector length n.

The false negative probability P fn is usually less constrained than P fp . In many applications, watermarking is a dissuasive weapon: P fn should be small enough that attackers don't take the risk of pirating content. Indeed, P fn ≈ 1/2 might be dissuasive enough. In the asymptotical setup, having E fn = 0 means that P fn is not converging to zero exponentially fast. It might converge to zero more slowly or it might converge to another value.

Note that once the watermark detector operates at a fixed E fp , say E fp = E, E fn depends on parameters (σ 2 X , σ 2 Z , P, E). The above assumptions on obliviousness imply that there is no guarantee about E fn at the detection side. However, just knowing that E fn > 0 even if it is by a very small amount, would prove that the dissuasion is achieved. This motivates the following definition. Definition 3.1. For a given setup (σ 2 X , σ 2 Z , P, E), a watermarking scheme operating at E fp = E is deemed robust if E fn > 0. We suppose that E fn is always a decreasing function w.r.t. σ 2 Z . The robustness R(σ 2 X , P, E) is the maximum noise power for which the watermarking scheme is robust. It is defined as

R(σ 2 X , P, E) := sup{σ 2 Z |E fn (σ 2 X , σ 2 Z , P, E) > 0}. ( 4
)
For a given setup (σ 2 X , σ 2 Z , P), the characteristic E fn = F (E fp ) is a decreasing function, illustrating the trade-off between the false negative and false positive probabilities. Usually, this characteristic vanishes to zero at some point that we name the right endpoint. Definition 3.2. The right endpoint of the characteristic is the biggest false positive error exponent for which the watermark is robust.

E R fp (σ 2 X , σ 2 Z , P) := sup{E|F (E) > 0}. ( 5
)

Prior art

The issue of obliviousness w.r.t. (σ 2 X , σ 2 Z ) at the detection side has been solved in two ways in the literature.

The first approach relies on Voronoï modulation (a.k.a. modulo channel) [START_REF] Zamir | Lattice Coding for Signals and Networks: A Structured Coding Approach to Quantization, Modulation and Multiuser Information Theory[END_REF]. Lattices embedding have been widely studied for decoding hidden messages (often called Quanitzed Index Modulation) [START_REF] Chen | Quantization index modulation: a class of provably good methods for digital watermarking and information embedding[END_REF][START_REF] Liu | On error exponents of modulo lattice additive noise channels[END_REF][START_REF] Moulin | Data-Hiding Codes[END_REF] but also in detecting zero-bit watermarking [START_REF] Liu | Error exponents for one-bit watermarking[END_REF]. It uses a Euclidean lattice Λ and the corresponding modulo operator (ì x mod Λ) to fold the space R n onto the Voronoi cell of that lattice.

In a nutshell, the fine grain (a.k.a. high resolution) assumption states that if the typical scale of the lattice is small compared to

σ 2 X + σ 2 Z , then ( ì X + ì Z mod Λ) is uniformly distributed over the
Voronoï cell of Λ. This can be also achieved thanks to a dithering signal which randomly shifts the lattice. In the end, the modulo operator succeeds to transform the unknown distribution of ì X + ì Z (because the detector is oblivious w.r.t. (σ 2 X , σ 2 Z )) into a known distribution (uniformity over the Voronoï cell). This in turn allows to compute and guarantee probability P fp .

The second approach uses a detection region W (2) which is a linear cone: If ì

x ∈ W, then α ì x ∈ W, ∀α > 0. This provides an invariance to scaling. If the distribution of ì X + ì Z is isotropic (as assumed in the theoretical setup), then ( ì

X + ì Z )/∥ ì X + ì
Z ∥ has a uniform distribution over the unit hypersphere. Again, this allows to compute and guarantee probability P fp .

The well-known dual hypercone detection is an example of this second approach with a score function defined as

s(ì x) = | ì x ⊤ ì u|/∥ ì x ∥, (6)
where ì u ∈ R n , ∥ ì u ∥ = 1, plays the role of a secret key. Threshold τ in ( 2) is defined as τ = cos(θ ). Region W is then the circular dual hypercone of axis ì u and semi-angle θ ∈ [0, π /2]. This scheme has a long tradition in the history of digital watermarking. Since the seminal papers of I. Cox et al. [START_REF] Cox | Secure spread spectrum watermarking for multimedia[END_REF][START_REF] Cox | Watermarking as communications with side information[END_REF], normalized correlation has been used in a vast majority of papers [START_REF] Furon | Broken arrows[END_REF] until sideinformation schemes were introduced [START_REF] Chen | Quantization index modulation: a class of provably good methods for digital watermarking and information embedding[END_REF][START_REF] Cox | Watermarking as communications with side information[END_REF]. The argument of the seminal paper [START_REF] Cox | Secure spread spectrum watermarking for multimedia[END_REF] was purely image processing oriented: normalizing the correlation is a way to be robust to contrast enhancement. Then some signal processing arguments defended this option [6, Sect. VI][1, Chap. 6, p. 237]: Decompose ì R as ( ì

R ⊤ ì u)ì u + ì R ⊥ where ì R ⊥ is the Euclidean projection of ì
R onto the subspace orthogonal to ì u. Under hypotheses H 0 and H 1 , this projection has the same distribution N ( ì

0 n-1 , N .I n-1 ). Variance N is then estimated by ∥ ì R ⊥ ∥ 2 /n-1
and used for comparing ì R ⊤ ì u to the threshold τ = N Φ -1 (1 -P fp ).

This indeed amounts to compare the ratio ì

R ⊤ ì u/∥ ì R ⊥ ∥ to a threshold, say 1 /tan(θ ), or equivalently, to compare s( ì R) = ì R ⊤ ì u /∥ ì R ∥ to cos(θ ).
Ten years later, Merhav et al. show that this scheme is optimal from the information theoretical veiwpoint [START_REF] Sabbag | Optimal Watermark Embedding and Detection Strategies Under Limited Detection Resources[END_REF]. Here is a brief summary of results concerning the dual hypercone in the literature [START_REF] Comesana | Asymptotically Optimum Universal Watermark Embedding and Detection in the High-SNR Regime[END_REF][START_REF] Sabbag | Optimal Watermark Embedding and Detection Strategies Under Limited Detection Resources[END_REF]:

E fp = -log sin(θ ), (7)
lim

σ 2 Z →0 E fn = 0 if A < cos(θ ) S A 2 cos 2 (θ ) otherwise ( 8
)
where

A := P/σ 2 X , (9)
S(x) := (x -1 -log(x))/2, ∀x ∈ R >0 . (10)
Note that function S(•) has a unique global minimum 0 at x = 1. One can see that the characteristic E fn = F (E fp ) is given by a parametric equation on θ . Usually, the watermarking power P is smaller than σ 2 X , so that A < 1. The right endpoint is then

E R fp = -1 /2 log(1 -A 2 ). (11)
Unfortunately, this characteristic is only known for σ 2 Z → 0. This is the reason why the authors of [START_REF] Comesana | Asymptotically Optimum Universal Watermark Embedding and Detection in the High-SNR Regime[END_REF] speak about 'high SNR regime'. Since it is a zero-order expression for σ 2 Z → 0, this is indeed the characteristic in the noiseless scenario. Our main contribution provides new results in the noisy scenario, i.e. when σ 2 Z > 0.

REVISITING THE NOISELESS SETUP

Before dealing with the noisy scenario, this section shows some hints about the noiseless scenario. Let us define the embeddable region as follows:

E(P) := { ì x ∈ R n |∃ì y ∈ W : ∥ ì x -ì y ∥ 2 < nP }. ( 12
)
This is the set of vectors in R n which can be successfully watermarked with a power budget P. This region is the filtering of W by a ball of radius √ nP, a.k.a. the result of the rolling ball technique [START_REF] Seeger | Smoothing a nondifferentiable convex function: the technique of the rolling ball[END_REF]: By rolling a ball of that radius over the boundary of W, the center of that ball draws the boundary of region E(P).

The main idea of this section is to note that, under the noiseless scenario, a false negative happens at the detection side whenever the embedding fails watermarking a given signal. Therefore,

P fn = P( ì X E(P)) = 1 -P( ì X ∈ E(P)). (13)
We are thus looking for a region W s.t. P( ì X ∈ W) = P fp and which, once filtered by the rolling ball technique, gives the lowest P fn , i.e. the biggest probability P( ì X ∈ E(P)). This is an elegant way to theoretically study side-informed watermarking under the noiseless scenario because there is no need to specify anything about the embedding mechanism (i.e. function ì w(•)).

Lower bound

The Gaussian isoperimetric inequality [START_REF] Ledoux | A Short Proof of the Gaussian Isoperimetric Inequality[END_REF] gives the worse possible region: For any region W ⊂ R n and ì

X ∼ N ( ì 0 n , σ 2 X I n ) s.t. P( ì X ∈ W) = P fp , we have P( ì X ∈ E(P)) ≥ Φ Φ -1 (P fp ) + A √ n , (14)
P fn ≤ 1 -Φ Φ -1 (P fp ) + A √ n . ( 15
)
Function Φ(•) is the cumulative density function of N (0, 1). According to the Gaussian isoperimetric theorem, equality happens if and only if W is a half-space. Following the definition (2) of W, this means that s(ì x) = ì x ⊤ ì u. The upper bound [START_REF] Shannon | Probability of error for optimal codes in a Gaussian channel[END_REF] translates into the following lower bound for E fn :

E fn ≥ A √ 2 -E fp + 2 . ( 16
)
In the same way, E R fp ≥ A 2 /2. This shows that a linear score function is indeed the worse choice in the noiseless setup, independently from the assumptions about obliviousness. Ironically, linear correlation was quite popular in the early ages of watermarking.

Dual hypercone

This section presents the methodology for calculating E fn with the rolling ball technique over the dual hypercone. Suppose that ì u = (1, 0, . . . , 0) ⊤ , without loss of generality. The definition of the embeddable region writes as:

E(P) =        ì x ∈ R n | n i=2 x 2 i ≤ |x 1 | tan(θ ) + √ nP cos -1 (θ )        . (17) Define V 1 = |X 1 |/ √ n and V 2 = n i=2 X 2 i /n.
These are two χ k random variables whose pdf is given by

f (v) = 2 √ n 2 k /2 Γ(k/2) v √ n σ X (k -1) e -n v 2 2σ 2 X (18)
with degree of freedom k 1 = 1 and k 2 = n -1 respectively. This change of variable yields a definition of set E(P) independent of n:

V = {(v 1 , v 2 ) ∈ R 2 ≥0 |v 2 ≤ v 1 tan(θ ) + √ P cos -1 (θ )}, (19)
so that

P fn = ∫ V f V 1 (V 1 )f V 2 (v 2 )dv 1 dv 2 (20) = K n ∫ V д(v 1 , v 2 )e -nh(v 1 ,v 2 ) dv 1 dv 2 , (21)
for some functions h(•) and д(•), and a multiplicative constant K n defined in the appendix. On one hand, we compute the 'exponent' of the multiplicative constant: κ := lim n→∞ -n -1 log K n . On the other hand, the Laplace method states that, as n → ∞, the integral is dominated by the value e -nh ⋆ where h ⋆ is the minimum of function h(•) over V (under some mild conditions). Then, the error exponent is given by:

E fn = min V h(v 1 , v 2 ) + κ. ( 22
)
For the dual hypercone in the noiseless scenario, calculations lead to:

E fn = min V v 2 1 2σ 2 X + S v 2 2 σ 2 X . (23)
This function has a unique global minimum 0 at v 1 = 0 and v 2 = σ X (see [START_REF] Liu | On error exponents of modulo lattice additive noise channels[END_REF]). This minimum lies in V if A ≤ cos(θ ). Otherwise, the solution of (22) lies on the boundary, i.e. v 2 = v 1 tan(θ ) + A cos -1 (θ ). This yields a univariate function in v 1 to be minimised, whose derivative takes only positive values. This shows that the minimum happens for the smallest value of v 1 , i.e. v 1 = 0 so that v 2 = A/cos(θ ) (see Fig. 2). This rediscovers the results (8) of [START_REF] Comesana | Asymptotically Optimum Universal Watermark Embedding and Detection in the High-SNR Regime[END_REF].

Interpretation: Probability P fn is dominated by the probability that ì X lies around the closest point to the origin in E(P). If this minimum distance √ nP cos(θ ) is lower than the typical module of ì X , i.e. nσ 2 X , then watermarking fails almost surely as n → ∞ (by concentration) so that E fn = 0.

THE NOISY SETUP

In the previous section, the rolling ball technique frees us to specify the way host vectors are watermarked. This section is now a little bit more specific for the hypercone detector. It is well known that ì w(ì x) must lie in the 2D subspace spanned by ì u and ì x [START_REF] Miller | Informed embedding: exploiting image and detector information during watermark insertion[END_REF][START_REF] Sabbag | Optimal Watermark Embedding and Detection Strategies Under Limited Detection Resources[END_REF]. We work on the following basis of this subspace:

ì e 1 = ì u, ì e 2 = (ì x -(ì x ⊤ ì u)ì u)/∥(ì x -(ì x ⊤ ì u)ì u)∥. ( 24
)
The watermark signal is crafted as:

ì w(ì x) = √ n( w1 ì e 1 + w2 ì e 2 ). (25)
Any other embedding strategy wastes embedding energy in space directions not useful for detecting the watermark. For the moment, ( w1 , w2 ) is not specified as a function of ì x. The distortion constraint imposes that w2 1 + w2 2 ≤ P. The appendix A shows with the same kind of change of variables and the use of the Laplace method (as in Sect. 4) that:

E fn = min F v 2 1 2σ 2 X + S v 2 2 σ 2 X + v 2 4 + v 2 5 2σ 2 Z + S v 2 3 σ 2 Z , (26)
with

F = {(v 1 + w1 + v 5 ) 2 tan 2 (θ ) ≤ (v 2 + w2 + v 4 ) 2 + v 2 3 } (27)
It is a priori not easy to solve this problem, but it is much simpler to see whether E fn = 0. This can only happen for v 1 = v 4 = v 5 = 0, v 2 = σ X , and v 3 = σ Z . Therefore, E fn = 0 if this point lies inside the feasible set F . This holds if and only if

H ( w1 , w2 ) ≤ σ 2 Z , with (28)
H ( w1 , w2

) := w2 1 tan 2 (θ ) -(σ X + w2 ) 2 . ( 29
)
Interpretation: Asymptotically, the performance of the scheme is governed by the way the typical realization of a host signal is watermarked. This typical host is orthogonal to the axis of the hypercone (v 1 = 0) and has norm

√ nσ X (because v 2 1 + v 2 2 = σ 2 X ). The typical realization of the noise has a norm √ nσ Z (because v 2 3 + v 2 4 + v 2 5 = σ 2 Z )
, is orthogonal to the axis (v 4 = 0) and is orthogonal to the host (v 5 = 0). E fn is null if this typical noise drives the watermarked signal outside the hypercone. The intersection of the hypercone with the plane v 3 = σ Z gives the hyperbola in R 2 :

C = {(a, b) ∈ R 2 |a 2 tan 2 (θ ) -b 2 = σ 2 Z }. ( 30
)

Provably good embeddings

This subsection assumes that the watermark designer has chosen a given dimension n. The requirement on the false positive rate imposes E fp ≈ E := -n -1 log(P fp ), which fixes the semi-angle θ by [START_REF] Furon | Broken arrows[END_REF]. What is the value of the robustness R(σ 2 X , P, E)? We adopt now the point of view of the embedder. Our goal is to avoid such a null error exponent E fn by carefully designing a watermark embedding ( w1 , w2 ). In a 2D plane mapping point ( w1 , w2 ), the embedding constraint w2 1 + w2 2 ≤ P defines a ball of radius √ P centered on (0, 0) whereas (28) defines a region delimited by an hyperbola (equality in (28)) of center (0, -σ X ). As σ Z → 0, the high-SNR regime tends to the noiseless scenario, and the hyperbola 'shrinks' towards its asymptotes: σ X + w2 = ± w1 tan θ . Figures 3,4, and 5 shows the situation.

When P is small, the entire ball is contained 'inside' the hyperbola (i.e. in between the two branches of the hyperbola as depicted in Fig. 3): Whatever the embedding ( w1 , w2 ), the false negative error exponent is zero. If P is big enough, the ball intersects the hyperbola and there are some embedding strategies ( w1 , w2 ) which provide non zero error exponent (Fig. 5). We are interested in the limit case when the ball has kissing points with the hyperbola (Fig. 4). The hyperbola is symmetric w.r.t. the axis { w1 = 0} and there are two kissing points (one on the

: P = σ 2 X cos 2 θ + σ 2
Z tan -2 θ . There are two kissing points given by (32) in green. Same setup as Fig. 3.

Interpretation. : Again, as R → 0 (i.e. in the noiseless setup), this E R fp tends to +∞ if P ≥ σ 2 X , or to -1 /2 log(1 -P /σ 2 X ) if P < σ 2 X . Expression (36) is thus compliant with [START_REF] Miller | Informed embedding: exploiting image and detector information during watermark insertion[END_REF].

As a special case, let R = σ 2 X , i.e. the maximum noise power equals the power of the host signal. Then, the necessary vector length to achieve both the requirement under H 0 (i.e. P fp ) and the target under H 1 (i.e. R = σ 2 X ) is (asymptotically):

n = 2 log P fp log 1 + P 2 /4σ 2 X -P /2σ 2 X = 4σ 2 X P (-log(P fp ) + o(1)). (37)
Interpretation. : This formula is interesting in audio watermarking, where the rule of thumb is that P/σ 2 X = cst, in the order of -20dB. This makes n in the order of some thousands.

COSTA'S BOUNDS

The previous section does not explicit the characteristic function E fn = F (E fp ) but focuses on its feature E R fn . This is sufficient for assessing whether the watermark is robust according to definition 3.1 while operating at E fp = E.

This section now compares the performances of the dual hypercone to bounds. It applies the idea of M. Costa in his famous paper [START_REF] Costa | Writing on dirty paper (Corresp.). Information Theory[END_REF] 1 :

• Non side-informed: The lower bound is given by removing the dependence on ì x in the definition of the watermark signal. Now, the watermark signal is a fixed vector:

ì w = √ nP ì u. The received vector is ì R = ì w + ì X + ì Z .
• Non blind: The upper bound comes by giving the detector an advantage: it knows the value of ì X . Removing this vector to received vector, we obtain ì R = ì w + ì Z .

In both cases, the detector takes a decision based on ì R = ì w + ì N , where ì N is a white Gaussian noise of variance N = σ 2 X + σ 2 Z (lower bound) or N = σ 2 Z (upper bound). We must not forget the specificities listed in Sect. 3.1. They make the detector oblivious to noise power N and watermark power P. This forbids the use of the Neyman-Pearson test s(ì x) = ì

x ⊤ ì u because fixing the threshold fulfilling the requirement on the probability of false positive P fp needs the knowledge of N . Again, we propose to use the scale invariant normalized correlation: s(ì x) = ì

x ⊤ ì u/∥ ì x ∥. Note that this time the detection region is a single circular hypercone due to the absolute value operator missing.

C. E. Shannon already tackled the study of the probability P fn in this case [START_REF] Shannon | Probability of error for optimal codes in a Gaussian channel[END_REF]. The error exponent is given by:

E fn = 0, if 0 ≤ θ ≤ θ 0 A 2 2 -A 2 G cos θ -log(G sin θ ) if θ 0 < θ ≤ π /2. ( 38
)
with A := P /N , tan(θ 0 ) = 1 /A, and G = (A cos(θ )+ A 2 cos 2 (θ ) + 4)/2. It follows that:

E R fp = 1 2 log 1 + P N , (39)
i.e. the capacity of a Gaussian channel. This is not a surprise. The semi-angle θ 0 defines the thinest cone for which the noise pushes the transmitted signal ì w outside with an exponentially vanishing 1 M. Costa used this idea for bounding the capacity of a side-informed communication channel. He never applied it to zero-bit watermarking, of course. probability P fn . This probability represents the decoding error probability in a communication scenario. As for P fp , it is the probability that a random vector falls into the hypercone under isotropic distribution. It equals the ratio of the solid angle of the hypercone and the one of the full hypersphere. E R fp can be though as representing, in logarithmic scale and per dimension, the number of hypercones with half angle θ 0 needed to fill the full hypersphere. In a communication scenario, this is the maximum number of messages (in logarithmic scale and per dimension) which can be reliably transmitted over this channel (i.e. with exponentially vanishing error probability).

The watermark is deemed robust 6 and7 shows that the dual hypercone scheme has performances very close to the lower bound. When analysed under criteria and setups making sense in digital watermarking, this scheme is deceiving. Note that the authors of [START_REF] Comesana | Asymptotically Optimum Universal Watermark Embedding and Detection in the High-SNR Regime[END_REF] claimed its optimality only in the noiseless scenario and under a so-called limited resources constraint.

if E R fp > E for N = σ 2 X + σ 2 Z ( lower
n =            2(-log(P fp )) log 1+ P /2σ

WATERMARKING REAL VECTORS

The above sections do not reveal how to watermark a host signal. They show that the error exponent E fn is governed by the way the typical host vector is watermarked. Asymptotically, this typical host vector is such that ì

x ⊤ ì u = 0 and ∥ ì x ∥ 2 = nσ 2 X because (v 1 , v 2 ) = (0, σ X ) in Sect. 5.1. In this case, the optimal watermark signal is given by (32). In practice, as n is not infinite, host vectors are different. How should they be watermarked?

The idea is to compute an error exponent E fn (ì x) dedicated for that host vector. We no longer rely on a statistical model of the host (i.e. white Gaussian noise). It amounts to replace random variables

(V 1 , V 2 ) in appendix A.1 by their occurences (v 1 , v 2 ) = (ì x ⊤ ì u, ∥ ì x ∥ 2 -ì x ⊤ ì u)/ √ n.
This modifies (42) to:

E fn (ì x) = min F( ì x ) v 2 4 + v 2 5 2σ 2 Z + S v 2 3 σ 2 Z , (42)
while F (ì x) has the same definition (27). Cancelling E fn (ì x) amounts to define a robustness level, whose maximisation defines the watermarking signal. This error exponent is null if and only if R(ì x, P, E) := max

w 2 1 + w 2 =P (v 1 + w1 ) 2 tan 2 (θ ) -(v 2 + w2 ) 2 ≤ σ 2 Z . ( 43
)
Note that this robustness depends on E through the semi-angle θ [START_REF] Furon | Broken arrows[END_REF]. Having a priori v 1 0 prevents finding a close form. The optimal embedding has to be found numerically with a line search prototyping ( w1 , w2 ) = √ P(cos β, sin β) for β ∈ [0, -π /2]. Figure 8 shows that the difference with (32) is small. This means that (32) is a good approximation of the optimal embedding.

Note that this derivation is not rigorous: on one hand the host is not an infinite vector, one the other hand we compute error exponent, i.e. asymptotical quantities. Yet, it justifies an embedding proposed by M. Miller, I. Cox and J. Bloom nineteen years ago [11, ) is shown as a vector starting at point (v 1 , v 2 ). They all have been scaled down to make the figure more visible. There is little difference in between the numerical solution of (43) in black and the suboptimal embedding (32) in magenta. The green line represents the boundary of the hypercone. The red line is the boundary of E(P) (the rolling ball smoothing of the detection region).

The color lines correspond to some level sets of R(ì x, P, E).

Eq. 4], well before that the concept of error exponent was introduced in digital watermarking. This paper theoretically confirms the remarkable intuition of this research team.

CONCLUSION

This paper completes the study of error exponents for the dual hypercone scheme. This scheme is important as it is one of the few schemes meeting the requirements on obliviousness at the embedding and detection sides. The paper extends the work of N. Merhav et al. to any SNR regime [START_REF] Comesana | Asymptotically Optimum Universal Watermark Embedding and Detection in the High-SNR Regime[END_REF][START_REF] Sabbag | Optimal Watermark Embedding and Detection Strategies Under Limited Detection Resources[END_REF]. It takes into account the specificities of digital watermarking. It introduces a new definition of robustness and the concept of necessary vector length to achieve false positive and robustness requirements.

In settings meaningful in digital watermarking, the asymptotical robustness is indeed not much bigger than the lower bound, i.e. the basic spread spectrum. This is in strong contrast with multibit watermarking, where M. Costa has shown that side-informed communication can perform as good as the upper bound. This is indeed due to the nature of zero-bit watermarking combined with the constraint of obliviousness. The open issue is whether there exist other detection regions granting obliviousness and closer to the upper bound. For instance, one can think of a union of thiner hypercones. Yet, their optimal number might be difficult to find.

Figure 3 :

 3 Figure 3: Conditions for E fn ≥ 0 in the plane ( w1 , w2 ): P is too small and E fn = 0. Setup: σ X = 1, σ Z = 0.5, and θ = π /3. The distortion constraint defines the red circle of radius √P centered on the origin (0, 0); E fn ≥ 0 defines the gray area outside the hyperbola of center C = (0, -σ X ) and of asymptotes the dashed blue lines.

Figure 4 :

 4 Figure 4: Conditions for E fn ≥ 0 in the plane ( w1 , w2 ): P = σ 2 X cos 2 θ + σ 2Z tan -2 θ . There are two kissing points given by (32) in green. Same setup as Fig.3.

Figure 6 :

 6 Figure 6: Robustness R(σ 2 X , P, E) as a function of E for σ 2 X = 1 and P = 0.1, i.e. function (33) and bounds (40) .

Figure 8 :

 8 Figure8: The watermark vector (w 1 , w 2 ) is shown as a vector starting at point (v 1 , v 2 ). They all have been scaled down to make the figure more visible. There is little difference in between the numerical solution of (43) in black and the suboptimal embedding (32) in magenta. The green line represents the boundary of the hypercone. The red line is the boundary of E(P) (the rolling ball smoothing of the detection region). The color lines correspond to some level sets of R(ì x, P, E).

  Necessary vector length n as a function of P /σ 2 X ∈ [0.01, 0.2], to achieve P fp = 10 -6 and R(σ 2 X , P, E) = σ 2 X , i.e. function (37) and bounds (41).

	10	4
											Hypercone
											Non side-informed
											Non blind
	10	3
	10 2	0	0.02	0.04	0.06	0.08	0.1	0.12	0.14	0.16	0.18	0.2
	Figure 7:

T. Furon left hand side, the other on the right hand side of the hyperbola -Fig. 4). The system of equations provided by (28) (with equality) and w2

1 + w2 2 = P implies that:

-(1 + tan 2 (θ )). w2 2 -2σ X . w2 + (P tan 2 (θ ) -σ 2 X -σ 2 Z ) = 0. (31)

This polynomial of degree two in w2 has a unique solution if and only if P = P 0 + σ 2 Z tan -2 (θ ) with P 0 := σ 2 X cos 2 (θ ). Consider the three following cases:

• if P ≤ P 0 then the ball and the hyperbola never intersect for any σ Z , including σ Z = 0 (as in Fig. 3): E fn = 0 for any σ Z . We rediscover result (8) from the noiseless scenario.

, then E fn = 0 for that particular noise power σ 2

Z , but it might be strictly positive for a less harmful attack.

Z tan -2 θ < P (as in Fig. 5), then E fn > 0 for this noise power and, in this sense, the watermark is robust to that attack.

When we have exact equality

Z tan -2 θ (as in Fig. 4), the two kissing points are given by (31):

A nice interpretation follows: if P = P 0 + δP with δP > 0, then w⋆2 1 = P 0 sin 2 θ + δP and w⋆2 2 = P 0 cos 2 θ . In words, the watermark signal first reaches the asymptotes of the hyperbola in order to guarantee E fn > 0 in the noiseless scenario. The 'shortest path' is to project (0, 0) on the asymptote by going along direction (sin θ,cos θ ). This consumes the embedding power P 0 . If it remains some extra embedding power δP > 0, the watermark signal carries on pushing the host signal only along the direction of the axis of the hypercone. This is depicted in Fig. 4.

The surprise is that this rediscovers the embedding shown to be 'optimum' in the noiseless scenario [START_REF] Comesana | Asymptotically Optimum Universal Watermark Embedding and Detection in the High-SNR Regime[END_REF]. In the noisy scenario, this embedding can indeed be deemed as optimal as well with the following meaning: It is not the embedding that maximizes E fn for a given σ 2 Z . This would certainly make ( w1 , w2 ) a function of σ 2 Z , which violated the obliviousness of the embedder. On the contrary, (32) is independent of σ 2 Z . It is the embedding that makes E fn > 0 over the biggest noise power range

). When this is the case, the robustness is increasing with P, but decreasing with σ 2 X . The robustness is also a decreasing function of E := (log P fp )/n: This scheme is extremely robustness for small E, i.e. very long signals. This complies with the well known rule of thumb in watermarking: The more spread, the more robust the watermark signal is. For a fixed P fp , and long signals, we have:

Figure 5: Conditions for E fn ≥ 0 in the plane ( w1 , w2 ): P is big enough so that there exist embeddings ( w1 , w2 ) on the red inside the gray area. Same setup as Fig. 3.

Maximum E fp

The previous section assumes that E fp = E, which fixes the semiangle θ of the hypercone by [START_REF] Furon | Broken arrows[END_REF], and establishes the expected robustness. This section provides another view: it assumes that σ 2 Z < R and looks for the biggest E fp for which E fn > 0, i.e. E R fp . For a given P fp , this gives a hint on the necessary vector length [15, Eq. ( 2) and below], n > (log(P fp ))/E R fp , to achieve a given robustness level. This analysis is done for the watermarking strategy (32), whenever applicable (i.e. if P > P 0 ). Inequality (28) implies that error exponent E fn is not null if:

A special case is R = 0 and P ≥ σ 2 X : The above inequality always holds which means that E fn is not null for any the angle of the hypercone θ ∈ (0, π /2], and thus for any value of E fp . This was shown in [START_REF] Ledoux | A Short Proof of the Gaussian Isoperimetric Inequality[END_REF] and [START_REF] Comesana | Asymptotically Optimum Universal Watermark Embedding and Detection in the High-SNR Regime[END_REF][START_REF] Sabbag | Optimal Watermark Embedding and Detection Strategies Under Limited Detection Resources[END_REF].

Yet in the noisy scenario, ( 35) is a polynomial of degree two w.r.t. ξ := sin 2 (θ ) which has two roots ξ -and ξ + s.t. ξ -< 0 < ξ + < 1 with:

This polynomial takes positive values outside the interval [ξ -, ξ + ]. This means that E fn > 0 if ξ + < sin 2 θ ≤ 1. This translates into the following right endpoint:

A LAPLACE METHOD IN THE NOISY SCENARIO A.1 Feasible set

Consider the basis (ì e 1 , . . . , ì e n ) of R n where the first two vectors are defined in (24). We introduce the following random variables:

V 1 , V 4 , and V 5 are Gaussian distributed while V 3 is a χ n-2 random variable scaled by σ Z / √ n. V 2 is a χ n-1 r.v. scaled by σ X (by the definition of ì e 2 , ì X ⊤ ì e 2 is always positive). With this formulation, a false negative happens if:

(49) This means that the random vector (V 1 , . . . , V 5 ) lies in the domain

A.2 Laplace potential functions

The random variables above defined are independent. The product of the their p.d.f. appears in the integral defining P fn . When rewriting this integral in the form (21), K n is thus the product of their multiplicative constants, whereas the potential function h(•) is the sum of their corresponding potential functions. This allows to deal case by case.

This gives K n = n /2πσ X whose exponent is κ = 0, and the potential function h(v) = -v 2 /2σ 2 X . A.2.2 Chi distribution with fixed degree. The p.d.f. is given in (18). Its multiplicative constant is K n = 2( n /2) k /2 /Γ( k /2)σ whose exponent is κ = 0. The potential function is h(v) = -v 2 /2σ 2 X . A.2.3 Chi distribution with increasing degree. The p.d.f. of V 3 for instance is (18) where k is a function of n: k = n-2. Its multiplicative constant is K n = 2( n /2) n-2 /2 /Γ( n-2 /2)σ Z , whose exposant is κ = -1/2. The potential function is h(v) = (v 2 /σ 2 Zlog(v 2 /σ 2 Z ))/2. In total, we get S(v 2 /σ 2 Z ).


MSc Zakarya Bentatou
email: zakarya.bentatou@etu.univ-amu.fr

MD Mathieu Finas
email: mfinas@chu-grenoble.fr

PhD Paul Habert
email: paul.habert@ap-hm.fr

MD; Frank Kober
email: frank.kober@univ-amu.fr

PhD Maxime Guye
email: maxime.guye@univ-amu.fr

MD PhD; Stéphanie Bricq
email: stephanie.bricq@u-bourgogne.fr

PhD Alain Lalande
email: alain.lalande@u-bourgogne.fr

PhD; Julien Frandon
email: julien.frandon@chu-nimes.fr

Jn Dacher

MD Benjamin Dubourg
email: benjamin.dubourg@chu-rouen.fr

MD; Jean-Nicolas Dacher
email: jean-nicolas.dacher@chu-rouen.fr

MD Gilbert Habib
email: gilbert.habib@ap-hm.fr

MD PhD; Jérome Caudron
email: drjeromecaudron@gmail.com

Md PhD; Sébastien Normant
email: sebastien.normant@chu-rouen.fr

BSc Stanislas Rapacchi
email: stanislas.rapacchi@univ-amu.fr

PhD Monique Bernard
email: monique.bernard@univ-amu.fr

PhD Alexis Jacquier
email: alexis.jacquier@ap-hm.fr

Aphm

Distribution of left ventricular trabeculation across age and gender in 140 healthy Caucasian subjects on MR imaging

Keywords: Myocardium, Left Ventricular Remodeling, Isolated Noncompaction of the Ventricular Myocardium, Left Ventricular Function, Healthy Volunteers balanced steady-state free precession C: compact myocardium mass CI: confidence interval DCM: dilated cardiomyopathy ECG: electrocardiogram ED: end-diastolic EDV: end-diastolic volume EF: ejection fraction HCM: hypertrophic cardiomyopathy ICC: intraclass correlation coefficient LV: left ventricle LVNC: left ventricular non-compaction MRI : magnetic resonance imaging SD: standard deviation T: trabeculated myocardium mass pg. 6

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Materials and methods

One hundred and forty healthy subjects were prospectively recruited and underwent cardiac MRI at 1.5T with a stack of short-axis cine sequences covering the entire LV. Enddiastolic volume (EDV), C and T masses were quantified using a semiautomatic method.

Ejection fraction (EF) and T/C ratio were computed.

Results

We included 70f/70m, mean age: 44 years±14, range: . The mean EF was 63.7±6.3%[50. .0], the mean EDV was 75.9±16.2ml/m 2 [36. 4-112.2], the mean C mass was 53.9±11.2g/m 2 [26. 5-93.4] and the mean T mass was 4.9±2.4g/m 2 [1. 1-11.4]. The T/C ratio was 9.2±4.5% [2.0-29.4]. Multivariate ANOVA test showed that the compact mass was influenced by EDV (P<0.0001), EF (P=0.001) and gender (P<0.0001), and the trabeculated mass depended on EDV (P<0.0001), gender (P=0.002) and age (P<0.0001), while the T/C ratio was only impacted by age (P=0.0003). Spearman test showed a correlation between EDV and C (r=0.60, P<0.0001), T (r=0.46, P<0.0001) and T/C ratio (r=0.26, P=0.0023).

Only T and T/C ratio were correlated to EF (r=-0.18, P=0.0373; r=-0.18, P=0.0321 respectively).

Conclusion

While the compact and trabeculated myocardium masses appear to relate separately to the cardiac function, age and gender, their ratio T/C appears to only decrease with age.

pg. [START_REF] Captur | prediction of sarcomere mutations in subclinical hypertrophic cardiomyopathy[END_REF] Furthermore, we propose here normal values for T, C and T/C in a cohort of healthy Caucasians subjects.

Running title: Distribution of LV trabeculations in healthy Caucasian subjects.

Introduction

Cardiac MRI has emerged as a reference modality for heart diseases investigation. It's a valuable non-invasive tool allowing an accurate morphological and functional characterization of the heart in several clinical conditions [START_REF] Sorensen | cardiac imaging: from fetus to adult[END_REF][START_REF] Legou | usefulness of t2 ratio in the diagnosis and prognosis of cardiac amyloidosis using cardiac mr imaging[END_REF].

Excess of trabeculation is a trait shared by several cardiomyopathies such as left ventricular non compaction (LVNC) [3]or hypertrophic cardiomyopathy (HCM) [START_REF] Captur | prediction of sarcomere mutations in subclinical hypertrophic cardiomyopathy[END_REF]. Excess of trabeculation was also described in pregnant women [START_REF] Gati | reversible de novo left ventricular trabeculations in pregnant women: implications for the diagnosis of left ventricular noncompaction in low-risk populations[END_REF] or highly trained athlete [START_REF] Gati | increased left ventricular trabeculation in highly trained athletes: do we need more stringent criteria for the diagnosis of left ventricular non-compaction in athletes?[END_REF].

Trabeculations are normal constituent of the left ventricle (LV) endocardial layer and discriminating between normal and excess of trabeculation is still a matter of debate.

Therefore, there is a need for reliable quantification of trabeculations in the LV myocardium and its variations in a healthy population. Indeed, to date, there is no criteria or method considered as gold-standard for differentiating between normal and pathological trabeculation of the myocardium. Recent semi-automatic method to quantify compact (C) and trabeculated (T) LV myocardium masses from cardiac magnetic resonance imaging (MRI) has been established and validated [7]. However, this method has yet to be evaluated in terms of the range of "normal" healthy values of LV masses in a large cohort.

The aim of this paper was to quantify the distribution of T and C myocardium masses across age and gender in a healthy population and to provide normal values for T, C and T/C ratio in healthy Caucasian subjects.

Materials and methods

Recruitment

This study was approved by the local Institutional Review Board in accordance with the guidelines outlined in the Declaration of Helsinki. One hundred and forty healthy volunteers were enrolled for a cardiac MRI examination after providing informed consent. Among them, forty-one were already included in an initial study by Bricq et al. [7]. An additional ninetynine healthy subjects (clinical trial NCT01712581, clinicaltrials.gov) were prospectively recruited from October 2012 to June 2014 and included in this cohort. A pregnancy test was performed for all female participants of childbearing age. Inclusion criteria were: age over 18 and below 70 years old, ability to understand and sign the consent form, ability to maintain supine position for 20 minutes, no cardiac symptoms, normal cardiovascular examination results and no known risk factors or history of cardiac disease. Exclusion criteria were: claustrophobia, transfused subjects, haematological disease, iron deficiency, high blood pressure, contraindication for MRI, pregnancy, diabetes and arrhythmias.

Image acquisition

Magnetic resonance (MR) examinations were performed on a 1.5 T Avanto ® scanner (Siemens Healthineers, Erlangen, Germany) using a multichannel body array coil combined with a spine array coil with the subjects in supine position. A stack of balanced steady-state free precession (bSSFP) cine sequences were acquired in short axis view in order to cover the whole LV from base to apex with a total of 7 to 14 slices. Acquisitions were electrocardiogram (ECG) synchronized and performed under inspiratory breath hold using the following typical parameters set: flip angle: 77°, pixel bandwidth: 930Hz, spatial resolution between 1.37x1.37mm 2 and 1.48x1.48mm 2 , slice thickness: 6mm, gap between slices: 9mm, pg. 8 repetition time: 3.62ms, echo time: 1.51ms, parallel imaging 2x (GRAPPA [START_REF] Griswold | generalized autocalibrating partially parallel acquisitions (grappa)[END_REF]). The resolution and the flip angle were adapted to subject morphology for each individual.

Image analysis

Argus ® Ventricular Function software (Siemens Healthineers, Erlangen, Germany) was used to quantify end diastolic volume (EDV), end systolic volume (ESV), end diastolic LV mass and to compute the ejection fraction (EF) for all subjects. The end-diastolic (ED) image was defined as the frame with the largest blood pool volume in the LV cavity. The endsystolic (ES) image was determined as the frame with the smallest blood pool volume in the LV cavity. The most basal slice of the LV was defined as the slice showing at least 50 % of the LV myocardium circumference at ED. The most basal section could differ between ED and ES.

LV trabeculations, endocardial and epicardial borders were defined, in the whole stack of the diastolic phase, using the semi-automatic method proposed by Bricq et al. [7] (Fig1).

Compact (C) and trabeculated (T) myocardium masses were defined as the difference between the epicardium and the endocardium and between the trabeculation contour and the endocardium respectively, as described in [7]. Papillary muscles were included in the compact LV myocardium mass. C and T masses were indexed to the body surface area (BSA) to take into account variation of height and weight between subjects. The T/C ratio was then computed and expressed as a percentage. All left ventricular parameters, C and T measurements were done by an operator (Op-1, three-year experience, Image Processing Engineer). A second operator (Op-2, two-year experience, Doctor) repeated the measurements on 10 randomly assigned subjects to evaluate inter-observer variability. To assess intraobserver variability, measurements were repeated by the first operator on 20 randomly selected subjects, at least 3 weeks later. All operators were blinded to the subject characteristics (age and gender) and to the other operator measurements. A representation of T/C ratio mean values across age and gender was produced for the 16 LV segments.

Statistical analysis

Statistical analysis was performed using GraphPad Prism version 7.0a (GraphPad software Inc., San Diego, CA). The intraclass correlation coefficient (ICC) and Bland-Altman method comparison were used to evaluate the reproducibility between two segmentations by Op-1 and between segmentations by Op-1 and Op-2. Values were reported as mean ± standard deviation (SD). The mean values were calculated according to gender (70 males and 70 females) and age (5 groups of 28 subjects, 14 men and 14 women). BSA was calculated using the formula of Dubois and Dubois [9]. D'Agostino & Pearson normality test was performed for the 5 age groups and each gender. For each variable (EF, EDV, ESV, T, C and T/C ratio), a non-parametric analysis (Mann-Whitney test for genders, Kruskal-Wallis test and Dunn multiple comparisons post-hoc test for age groups) was used to compare groups who did not pass the normality test. Groups who passed the normality test were analysed using a parametric test (Welch's t test for genders, ordinary one-way ANOVA with Tukey's multiple comparisons test for age groups). A Multivariate ANOVA (MANOVA) was performed to test the influence of gender and age on quantitative cardiac MRI measurements.

A Spearman test was also performed in order to measure the correlation between LV masses and the other LV parameters (EF and EDV). Significance was set to P<0.05. pg. 10

Results

Participant Characteristics

Volunteers (70 men and 70 women) were organised so as to include 14 men and 14 women in each of the 5 following groups: first group, age between 20-29 years old; second group, age between 30-39 years old; third group, age between 40-49 years old; fourth group, age between 50-59 years old and the fifth group, age between 60-69 years old. Subject characteristics were summarized in Table 1. All quantitative variables were expressed as mean±SD and ranges. There was no significant difference (P>0.05) between any age group in terms of height, weight and BSA. Men had significantly greater BSA than women (P<0.0001).

Left ventricular parameters

Regarding the ejection fraction (EF), all age groups were similar, but there was a significant difference (P=0.0178) between men (62.4±6.1 %) and women (64.9±6.2 %) as shown in Table 2. EF increased significantly with age (P=0.0184) (Fig. 2a). The mean value of end-diastolic volume (EDV) was significantly greater in men (78.8±18.0 ml/m 2 ) than in women (72.8±13.6 ml/m 2 , P=0.0280). When comparing EDV across age groups, it decreased significantly with age from 87.13±16.74 ml/m 2 in the youngest to 67.37±17.09 ml/m 2 in the oldest (P<0.0001). ESV also decreased significantly with age (P<0.0001) (Fig. 2b) with significant difference between men and women (P=0.0052) and the age group 20-29 and 40-49 (P=0.0005), 50-59 (P=0.0017) and 60-69 (P<0.0001).

Left ventricular masses across gender and age groups

There was no significant difference between age groups in terms of compact myocardium mass. The mean value of C was significantly greater in men (60.0±9.1 g/m 2 ) than in women (47.7±9.8 g/m 2 , P<0.0001) (Table 2). While on the other hand, C remained stable across age (Fig. 2c).

pg. 11

Trabeculated myocardium mass decreased significantly with age from 6.85±2.56 g/m 2 in the first age group 20-29 to 3.97±1.99 g/m 2 in the oldest (fifth age group: 60-69) (P<0.0001) (Fig. 2c) with significant difference between the first group (age between 20-29) and all the other age groups (30-39, P=0.0072; 40-49, P=0.0025; 50-59, P=0.0001; and 60-69, P<0.0001).

The difference was also significant between genders with greater values for men: 5.4±2.3 g/m 2 , compared to women: 4.0±2.3 g/m 2 (P=0.0052) (Table 2).

T/C ratios were not significantly different between genders (9.1±3.7 % for men and 9.2±5.2 % for women). The T/C ratio was decreasing with age from 12.33±4.57 % in the youngest to 7.55± 3.49 % in the oldest (P=0.0003) (Fig. 2d). There was also a significant difference between the first age group 20-29 and the age groups 40-49 (P=0.0390), 50-59 (P=0.0009) and 60-69 (P=0.0006).

Values of the T/C ratio for men and women in each age group are listed in Table 3, as mean±SD and 95% confidence interval (CI).

Observer variability

Mean and SD of T, C and T/C ratio for the intra-observer variability, are provided in Table 4. The ICC (Table 4, ICC) was high (>0.90) for all parameters (T, C and T/C ratio).

The Bland-Altman bias between the segmentations was -0.16±0.39 g for T mass, 0.70±2.00 g for C mass and -0.21±0.47 % for the T/C ratio. Table 4 displays also the variability between the two experts Op-1 and Op-2 where the ICC was computed. We had a high ICC (³0.80) for C, T and T/C ratio. Bland-Altman bias was found to be -0.64±0.39 g for the T mass, -3.61±6.04 g for the compact myocardium mass and -0.58±1.81 % for the T/C ratio. The mean values ± SD for the three parameters (T, C, and T/C ratio) are also given in Table 4.

pg. 12

Relationship between LV masses and age, gender and LV parameters

In terms of trabeculated myocardium mass, there was a significant correlation with age, gender and EDV, but not with EF. The only factor independently linked with the T/C ratio variation was age (Table 5).

The Spearman test showed a positive correlation between EDV and C, T and T/C ratio (r=0.60, P<0.0001; r=0.46, P<0.0001; r=0.26, P=0.0023 respectively), but only T and T/C ratio were negatively correlated to EF (r=-0.18, P=0.0373; r=-0.18, P=0.0321 respectively).

Values of the T/C ratio across the 16 LV segments

The distribution of the highest T/C ratios (over 20 %) was found in apical anterior and apical lateral segments. T/C ratios in the basal third did not reach 10 % for any subject regardless of gender and age as shown in Fig. 3.

pg. 13

Discussion

The main findings of this study were: 1) Local distribution of trabeculations showed that T/C ratios vary substantially across the LV, with higher ratios in apical anterior and apical lateral segments. 2) The T/C ratio was inversely correlated with age and independent to LV volumetric parameters in that cohort.

Trabeculations are normal constituent of LV and discrimination between normal trabeculation and hypertrabeculation remains a matter of debate [START_REF] Arbustini | left ventricular noncompaction[END_REF]. The presence of hypertrabeculation has been linked with a response to mechanical load which may disappear as the mechanical load dissipates [START_REF] Gati | reversible de novo left ventricular trabeculations in pregnant women: implications for the diagnosis of left ventricular noncompaction in low-risk populations[END_REF][START_REF] Gati | increased left ventricular trabeculation in highly trained athletes: do we need more stringent criteria for the diagnosis of left ventricular non-compaction in athletes?[END_REF]. Therefore, the evocation of hypertrabeculation might have important consequence in highly trained athletes. Furthermore, hypertrabeculation was diagnosed in asymptomatic subjects with normal or nearly normal ejection fraction [START_REF] Habib | isolated left ventricular non-compaction in adults: clinical and echocardiographic features in 105 patients. results from a french registry[END_REF].

Criteria based on segmental analysis using ultrasound (US), used to diagnose hypertrabeculation, are poorly correlated and are prone to confounding factors [START_REF] Kohli | diagnosis of left-ventricular non-compaction in patients with left-ventricular systolic dysfunction: time for a reappraisal of diagnostic criteria?[END_REF]. Kohli et al. showed that segmental measurements of T and C thickness using US are too sensitive to partial volume effect and slice selection. Authors concluded that those criteria result in over diagnosis of LVNC. Using MRI, Kawel et al. showed in 323 healthy subjects without cardiac disease or hypertension that more than 40% of subjects presented a trabeculated over compact myocardium thickness ratio higher than 2.3 [START_REF] Kawel | trabeculated (noncompacted) and compact myocardium in adults: the multi-ethnic study of atherosclerosis[END_REF]. However, the thickness ratio was assessed in only one segment in that study. The present study confirmed that LV trabeculations are not homogeneously distributed over LV cavity and a segmental measurement of trabeculation might reflect this variation. André et al. [START_REF] Andr | reference values for left and right ventricular trabeculation and non-compacted myocardium[END_REF] showed similar result using MRI in 117 healthy subject using only the long axis view. Quantification of the total amount of LV trabeculation appears preferable to assess normal LV trabeculation and already showed ability to discriminate between normal and diseased patients [START_REF] Jacquier | measurement of trabeculated left ventricular mass using cardiac magnetic resonance imaging in the diagnosis of left ventricular non-compaction[END_REF]. Our results were in line with the current literature in which the T/C ratio varies from 9.7 to 12.0 % [START_REF] Jacquier | measurement of trabeculated left ventricular mass using cardiac magnetic resonance imaging in the diagnosis of left ventricular non-compaction[END_REF][START_REF] Papavassiliu | effect of endocardial trabeculae on left ventricular measurements and measurement reproducibility at cardiovascular mr imaging[END_REF][START_REF] Grothoff | value of cardiovascular mr in diagnosing left ventricular non-compaction cardiomyopathy and in discriminating between other cardiomyopathies[END_REF]. Papavassiliu et al. pg. 14 found that trabeculae account for 10 % of the left ventricular mass [START_REF] Papavassiliu | effect of endocardial trabeculae on left ventricular measurements and measurement reproducibility at cardiovascular mr imaging[END_REF], but only 20 healthy volunteers were included using a fast-gradient echo sequence, whereas bSSFP sequences with higher resolution would be beneficial [START_REF] Alfakih | normal human left and right ventricular dimensions for mri as assessed by turbo gradient echo and steady-state free precession imaging sequences[END_REF]. Using a manual segmentation and bSSFP sequences, Jacquier et al. found a T/C ratio of 12±5 % for healthy subjects [START_REF] Jacquier | measurement of trabeculated left ventricular mass using cardiac magnetic resonance imaging in the diagnosis of left ventricular non-compaction[END_REF]. Grothoff et al. assessed the T/C ratio by manual contouring in 24 healthy volunteers and found a mean value of 9.9±4.4 % [START_REF] Grothoff | value of cardiovascular mr in diagnosing left ventricular non-compaction cardiomyopathy and in discriminating between other cardiomyopathies[END_REF].

Obtaining normal values and ranges for trabeculated mass and T/C ratio is crucial as trabeculated myocardium is a determinant phenotypic expression in several cardiac diseases.

In an early study, Kozor et al. showed that, in Fabry disease and in HCM, both relative papillary muscle mass and T mass values were increased [START_REF] Kozor | a disproportionate contribution of papillary muscles and trabeculations to total left ventricular mass makes choice of cardiovascular magnetic resonance analysis technique critical in fabry disease[END_REF]. In contrast, these authors also showed that neither in hypertrophy linked to afterload increase (aortic stenosis and hypertension) nor in amyloidosis were there increased trabeculations. Furthermore, Choudhary suggested that a higher T/C ratio was associated with a higher prevalence of electrical disturbances including sustained ventricular tachycardia in patients suffering from LVNC [START_REF] Choudhary | improving the diagnosis of lv non-compaction with cardiac magnetic resonance imaging[END_REF]. Therefore a precise quantification of LV trabeculation might be clinically relevant to discriminate between normal and excess of trabeculation in specific cardiomyopathies, despite negative findings in the follow-up of healthy subjects [START_REF] Zemrak | the relationship of left ventricular trabeculation to ventricular function and structure over a 9.5-year follow-up[END_REF].

An additional finding of our study is that T mass and T/C ratio decreased with age, with no correlation with either LV volumes, nor EF. The compact myocardium mass remained stable across age. This might be due to the physiological process of aging. This result was in line with findings from André et al. who also showed a decrease of LV trabeculated mass with age [START_REF] Andr | reference values for left and right ventricular trabeculation and non-compacted myocardium[END_REF]. Dawson et al. suggested a decrease in the T/C thickness ratio with age, using the T/C thickness ratio measured in the end systolic phase with MRI [START_REF] Dawson | regional thicknesses and thickening of compacted and trabeculated myocardial layers of the normal left ventricle studied by cardiovascular magnetic resonance[END_REF], and this trend was confirmed in our study. Lastly, no gender differences have been reported in the literature regarding T/C ratio [7,[START_REF] Dawson | regional thicknesses and thickening of compacted and trabeculated myocardial layers of the normal left ventricle studied by cardiovascular magnetic resonance[END_REF], which is in accordance with our results. The T/C ratio has been pg. [START_REF] Jacquier | measurement of trabeculated left ventricular mass using cardiac magnetic resonance imaging in the diagnosis of left ventricular non-compaction[END_REF] shown to be different between ethnic groups and seems to be higher in Chinese and African-American populations [START_REF] Kawel | trabeculated (noncompacted) and compact myocardium in adults: the multi-ethnic study of atherosclerosis[END_REF], therefore our values may be only applicable in Caucasian populations.

The non-compaction of the myocardium has been studied in a large cohort of patients in the MESA study, but was limited to the measurement of the thickness of the compact and trabeculated LV wall [START_REF] Zemrak | the relationship of left ventricular trabeculation to ventricular function and structure over a 9.5-year follow-up[END_REF] or using the global fractal approach [23]. The proposed segmentation tool integrates other crucial functional parameters like EDV and LV wall mass which could help in patients stratification. Indeed, some of these parameters are prone to affect the morphology and amount of LV trabeculations [START_REF] Kawel | trabeculated (noncompacted) and compact myocardium in adults: the multi-ethnic study of atherosclerosis[END_REF][START_REF] Andr | reference values for left and right ventricular trabeculation and non-compacted myocardium[END_REF][START_REF] Choudhary | improving the diagnosis of lv non-compaction with cardiac magnetic resonance imaging[END_REF].

Our method has yet to be applied in a larger population, prospectively, using a multicentric design. In our study, only one ethnic group was included.

We have proposed normal values for T, C and T/C mass ratio in a cohort of healthy Caucasian subjects. These normal values of the T/C ratio are 9.1±3.7 % for men and 9.2±5.2 % for women; within each gender, T/C decreases with greater age. While the compact and trabeculated myocardium masses appear to relate to both cardiac function and gender, there was no correlation between these parameters and the T/C ratio.

Table and figure legends

Figure 1 .

 1 Figure 1. Example of the 16-segments left ventricular segmentation over base (left), median

Figure 2 .

 2 Figure 2. Influence of age on left ventricular parameters and masses. (a) Distribution of

Figure 3 .

 3 Figure 3. Mean measurement values of the ratio T/C for the 16 LV segments. C, compact

Figure 1 .

 1 Figure 1. Example of the 16-segments left ventricular segmentation over base (left),

Figure 2 .

 2 Figure 2. Influence of age on left ventricular parameters and masses. (a) Distribution of ejection

Figure 3 .

 3 Figure 3. Mean measurement values of the ratio T/C for the 16 LV segments. C, compact

Table 1 .

 1 Baseline characteristics of the healthy volunteers (mean±SD & ranges)

Table 2 .

 2 Left ventricular parameters and masses (mean±SD & 95% CI).

Table 3 .

 3 Values of T/C ratio for men and women by age groups (mean±SD & 95% CI).

Table 4 .

 4 Intra-observer & inter-observer variability.

Table 5 .

 5 Impact of age, gender, ejection fraction and end-diastolic volume on left ventricular masses.

Table 3 .

 3 Intra-observer & inter-observer variability

	Intra-observer
	Measurement

Table 4 .

 4 Impact of age, gender, ejection fraction and end-diastolic volume on left ventricular masses

			Multivariate ANOVA
		Whole model	Age	Gender	EF	EDV
	C	p<0.0001	p=0.67	p<0.0001	p=0.001	p<0.0001
	T	p<0.0001	p=0.007	p=0.02	p=0.91	p<0.0001
	T/C ratio	p=0.0009	p=0.008	p=0.53	p=0.26	p=0.27
	C, compact myocardium mass; T, trabeculated myocardium mass; EF, ejection fraction; EDV,
	end-diastolic volume.

Acknowledgements

We would like to thank Dr. Ben Ridley for his help in editing the English version of the manuscript.

pg. 17

Funding:

This study has received funding by CHU-Hôpitaux de Rouen, Direction de la Recherche et de l'Innovation.

Conflict of interest:

The authors of this manuscript declare no relationships with any companies, whose products or services may be related to the subject matter of the article.

Statistics and biometry:

One of the authors has significant statistical expertise.

Informed consent:

Written informed consent was obtained from all subjects in this study.

Ethical approval:

Institutional Review Board approval was obtained.

Study subjects or cohorts overlap:

Some study subjects or cohorts have been previously reported in