Alexandre Seuret
email: aseuret@laas.fr

Frédéric Corentin Briat
email: briatc@bsse.ethz.ch

F Gouaisbaut

come

Stability analysis of asynchronous sampled-data systems with discrete-time constant input delay

Stability analysis of asynchronous sampled-data systems with discrete-time constant input delay

Alexandre Seuret, Corentin Briat and Frédéric Gouaisbaut

Abstract-Aperiodic sampled-data systems controlled by a delayed state-feedback controller arise naturally in networked control systems. We propose a novel way to analyze the stability of sampled-data systems with discrete-time constant input delay by, first, reformulating the system into an aperiodic discretetime system with constant delay and by, then, considering a mixture of a discrete-time Lyapunov-Krasovskii functional and a looped-functional in order to account for the presence of the delay and the sampling aperiodicity, respectively. Stability criteria, expressed as linear matrix inequalities, are derived for both certain and uncertain (time-varying) systems. An example is given for illustration.

I. INTRODUCTION

Sampled-data systems are a particular class of systems that have been extensively studied in the literature [START_REF] Chen | Optimal sampled-data control systems[END_REF] as they arise, for instance, in digital control [START_REF] Middleton | Digital Control and Estimation -A Unified Approach[END_REF] and networked control systems [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF], [START_REF] Zampieri | A survey of recent results in Networked Control Systems[END_REF]. Several types of aperiodic sampling is not as well understood as the periodic case since the nonstationarity of the sampling process leads to additional difficulties in the analysis problem. Several approaches have been proposed in this regard. Discretetime systems approaches have been discussed in [START_REF] Suh | Stability and stabilization of nonuniform sampling systems[END_REF]- [START_REF] Cloosterman | Stability of networked control systems with uncertain time-varying delays[END_REF] where the sampling-period-dependent exponential terms in the stability conditions are embedded in a convex polytope, and are therefore considered as time-varying uncertainties. This approach leads to efficient and tractable conditions that can be easily used to design controllers. However, the approach is limited by the fact that only certain and timeinvariant systems can be considered. Another approach is based on the so-called "input-delay approach" and consists of reformulating the original sampled-data system into a timedelay system [START_REF] Fridman | Robust sampled-data stabilization of linear systems: An input delay approach[END_REF]- [START_REF] Seuret | Stability analysis for sampled-data systems with a timevarying period[END_REF]. The system can then be analyzed using delay-techniques such as the Lyapunov-Krasovskii theorem. The main advantage of this approach lies in the possibility for considering uncertain, time-varying and even nonlinear systems. One limitation is the difficulty for designing controllers. Robust analysis techniques have also been applied in a successful way using small-gain results [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-and-hold circuits[END_REF], IQCs [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-andhold devices[END_REF]- [START_REF] Kao | On robust stability of aperiodic sampleddata systems -an integral quadratic constraint approach[END_REF] or well-posedness theory [START_REF] Ariba | Simple conditions for L 2 stability and stabilization of networked control systems[END_REF]. Approaches based on impulsive systems have also been considered using functionals [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF] or clock-dependent Lyapunov functions [START_REF] Briat | Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints[END_REF].

The latter approach has been shown to be able to both characterize the robust stability of periodic and aperiodic sampled-data systems in both the certain and (time-varying) uncertain cases. Interestingly, robust stabilization conditions for sampled-data systems can be easily formulated in a convex form using this approach. In this regard, this framework combines the efficiency of discrete-time and functional approaches. Finally, the looped-functional-based approach has been proposed in [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF]- [START_REF] Briat | Convex dwell-time characterizations for uncertain linear impulsive systems[END_REF] to address the stability analysis problem of sampled-data and impulsive systems. This particular type of functional has the interesting property of not being necessarily required to be positive definite. Instead of that, we demand the fulfillment of a "looping condition", a certain boundary condition that can be made structurally satisfied while constructing the functional. In this regard, this class of functionals is therefore more general than usual Lyapunov(-Krasovsksii) functionals as the loopingcondition turns out to be weaker than the positive definiteness condition; see e.g. [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF]- [START_REF] Briat | Convex dwell-time characterizations for uncertain linear impulsive systems[END_REF].

We propose to address the problem of stability analysis of (uncertain) aperiodic sampled-data systems with discretetime input delay. While the delayed sampled-data systems considered in [START_REF] Seuret | Stability analysis of networked control systems with asynchronous sampling and input delay[END_REF], [START_REF] Liu | Wirtinger's inequality and Lyapunov-based sampled-data stabilization[END_REF] are subject to a continuous-time delay (the delay is in seconds), the systems we are interested in here involve a discrete-time delay (the delay is a number of samples). A solution to this problem, based on state augmentation, has been proposed in [START_REF] Seuret | Stability analysis of uncertain sampled-data systems with incremental delay using looped-functionals[END_REF]. The approach has been shown to yield accurate results but with a quite high computational complexity preventing then the application of the method when the delay is large. This is the reason why we will focus here on an alternative approach which is not based on state augmentation. Instead, the corresponding aperiodic discrete-time system with delay is directly considered and analyzed using an adequate blend of Lyapunov-Krasovskii and looped-functionals. The approach is applied to both certain and (time-varying) uncertain systems for which stability criteria are derived and formulated as LMIs.

Outline. Section III states preliminary technical results for dealing with LMIs and looped-functionals. The main stability results addressing both the nominal and robust stability of sampled-data systems with delay are presented in Section IV. An illustrative example is finally discussed in Section V.

Notations. Throughout the paper, R n denotes the ndimensional Euclidean space endowed with the vector 2norm | • |. The space of n × m real matrices is denoted by R n×m . For a matrix P , the notation P ≻ 0 means that P is symmetric and positive definite. The sets S n and S n + are the set of n-dimensional symmetric and symmetric positive definite matrices, respectively.

II. PROBLEM FORMULATION

Let us consider here linear continuous-time systems of the form ẋ(t) = Ax(t) + Bu(t), t ≥ 0,

x(0) = x 0 (1)
where x ∈ R n and u ∈ R m are the state of the system and the control input, respectively. Above, the matrices A and B may be uncertain and/or time-varying. This will be explicitly mentioned when this is the case. The control input u obeys the following equation

u(t) = Kx(t k-h ), t ∈ [t k , t k+1 ), k ∈ N, (2)
where K ∈ R m×n is the controller gain and h ∈ N is the constant time-delay. The sequence {t k } k∈N , t 0 = 0, is the sequence of sampling instants that we assume to obey the condition

T k := t k+1 -t k ∈ [T min , T max ] for all k ∈ N
and for some 0 < T min ≤ T max < ∞. Note that this condition implies that the sequence is increasing and admits no accumulation point, i.e.

t k → ∞ as k → ∞.
The closed-loop system obtained from the interconnection of ( 1) and ( 2) is given by

ẋ(t) = Ax(t) + BKx(t k-h ), t ∈ [t k , t k+1 ), k ∈ N, x(t -θ ) = ψ(θ), θ = 0, . . . , h (3)
for some ψ(θ) ∈ R n with ψ(0) = x 0 and where {t -h , . . . , t -1 } is a sequence of negative real numbers satisfying t -h < . . . < t -1 < 0. Note that since only stability is considered, then ψ(θ) and {t -h , . . . , t -1 } can be arbitrarily chosen. Alternatively, the above system can be represented by the time-delay discrete-time system

x(t k+1 ) = A d (T k )x(t k ) + B d (T k )x(t k-h ), k ∈ N, x(t -θ ) = ψ(θ), θ = 0, . . . , h (4) where
A d (T k ) = e AT k and B d (T k ) = T k 0 e As BKds.
Remark 1: It is important to stress that the system (3) dramatically differs from the system

ẋ(t) = Ax(t) + BKx(t k -d), t ∈ [t k , t k+1 ) x(θ) = φ(θ), θ ∈ [-d, 0], (5)
defined for some delay d > 0, that has been studied, for instance, in [START_REF] Liu | Wirtinger's inequality and Lyapunov-based sampled-data stabilization[END_REF], [START_REF] Naghshtabrizi | Stability of delay impulsive systems with application to networked control systems[END_REF], [START_REF] Seuret | Stability analysis of networked control systems with asynchronous sampling and input delay[END_REF]. The delay here acts in a continuous way on the delayed term whereas in (3) the delays acts discretely. Note also that the above system is infinitedimensional while (3) is finite-dimensional. Both systems actually coincide whenever the sampling-period T and the delay h are constant and verify d = hT .

III. PRELIMINARIES

A. A preliminary result

The following result can be seen as a particular case of the reciprocally convex combination result derived in [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF]:

Lemma 3.1 ( [29]): Let R ∈ S n + and α ∈ (0, 1). Then, we have that R/α 0 * R/(1 -α) I -I ⊺ R I -I . ( 6
)
Proof: We present here an alternative proof to the one in [START_REF] Seuret | Integral inequality for time-varying delay systems[END_REF]. First note that we obviously have

(1/α -1)(R -R) 0
and, from the non-strict Schur complement [START_REF] Boyd | Linear Matrix Inequalities in Systems and Control Theory[END_REF] and the facts that α ∈ (0, 1) and R ≻ 0, we then obtain the equivalent condition

(1/α -1)R R R (1/(1 -α) -1)R 0
where we have used the identity (1/α-1)

-1 = 1/(1-α)-1.
This concludes the proof. Note that the above bound is computationally less complex than the original one derived in [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF] since no additional variable is introduced here. Regarding accuracy, it has been emphasized through several examples in [START_REF] Seuret | Integral inequality for time-varying delay systems[END_REF] that the inequality from [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF] and the one provided in Lemma 3.1 yield equivalent results for fast-varying delays systems.

B. Lifting of sampled-data systems trajectories

The looped-functional-based approach relies on the characterization of the trajectories of system (3) in a lifted domain [START_REF] Yamamoto | New approach to sampled-data control systems -a function space method[END_REF], [START_REF] Briat | A looped-functional approach for robust stability analysis of linear impulsive systems[END_REF]. We therefore view the entire state-trajectory as a sequence of functions {x(t k + τ ), τ ∈ (0, T k ]} k∈N with elements having a unique continuous extension to [0, T k ] defined as

χ k (τ ) := x(t k + τ ) with χ k (0) = lim s↓t k x(s).
Finally, we define K as

K := T ∈[Tmin, Tmax] C([0, T ] → R n ).
Using this notation, the system (3) rewrites as

χk (τ ) = Aχ k (τ ) + BKχ k-h (0), τ ∈ [0, T k ), k ∈ N. (7)
In what follows, we also consider the notation

χ h k (θ) = χ k+θ (0) = x(t k+θ ) where θ ∈ {-h, -h + 1, . . . , 0}. We also denote by χ h k the vector col(χ h k (0), χ h k (-1), . . . , χ h k (-h)).

C. Definition and results about looped-functionals

Let us first define looped-functionals Definition 3.2 ( [START_REF] Briat | A looped-functional approach for robust stability analysis of linear impulsive systems[END_REF]):

Let 0 < T min ≤ T max < ∞. A functional f : [0, T max ] × K × [T min , T max ] → R, is said to be a looped-functional if 1) the equality f (0, z, T ) = f (T, z, T ) (8)
holds for all functions z ∈ K and all T ∈ [T min , T max ] and 2) it is differentiable with respect to the first variable with the standard definition of the derivative. The set of all such functionals is denoted by LF([T min , T max ]).

The idea for proving stability of (3) is to look now for a positive definite quadratic form V such that the sequence {V (χ k (0), χ h k )} k∈N is monotonically decreasing. This is formalized below through a functional existence result:

Theorem 3.3: Let 0 < T min ≤ T max < ∞, h ∈ N and V : R n × R n(h+1) → R + be a quadratic form verifying µ 1 |ϕ 2 | 2 ≤ V (ϕ 1 , ϕ 2 ) ≤ µ 2 |ϕ 2 | 2 (9)
for some scalars 0 < µ 1 ≤ µ 2 and for any

(ϕ 1 , ϕ 2 ) ∈ R n × R n(h+1) where ϕ 2 = ϕ ⊺ 1 * ⊺ .
Assume further that one of the following equivalent statements hold:

1) The sequence {V (χ k (0), χ h k )} k∈N is decreasing. 2) There exists a looped-functional V ∈ LF([T min , T max ]) such that the functional W k defined as W k (τ, χ k , χ h k ) := τ T k Λ k + V (χ k (τ ), χ h k ) +V(τ, χ k , T k ) (10)
with

Λ k = V (χ k+1 (0), χ h k+1 ) -V (χ k (T k ), χ h k
) is decreasing along the trajectories of the system (7), i.e. the derivative

d dτ W k (τ, χ k , χ h k ) := 1 T k Λ k + d dτ V(τ, χ k , T k ) + d dτ V (χ k (τ ), χ h k ) (11) is negative definite for all τ ∈ [0, T k ], T k ∈ [T min , T max ], k ∈ N.
Then, the system (3) with delay h is asymptotically stable for any sequence {t k } k∈N satisfying t k+1 -t k ∈ [T min , T max ], k ∈ N.

Proof: The proof is similar to the one derived in [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF], [START_REF] Briat | A looped-functional approach for robust stability analysis of linear impulsive systems[END_REF] and is thus omitted.

IV. STABILITY ANALYSIS

A. Nominal stability analysis

This section provides a stability result for aperiodic sampled-data systems with a constant delay h. We have the following result:

Theorem 4.1: The sampled-data system [START_REF] Hetel | Stabilization of arbitrary switched linear systems with unknown time-varying delays[END_REF] with the delay h is asymptotically stable for all

T k := t k+1 -t k ∈ [T min , T max ], k ∈ N, if there exist matrices R, Q, Z ∈ S n + , P, X ∈ S 2n , S 1 , S 2 ∈ S n , U 1 , U 2 ∈ R n×n such that the LMIs Φ 0 := I I P I I ≻ 0, Φ 1 (θ) := F 0 (θ) + θF 1 θM ⊺ 0 Z ⋆ -Z ≺ 0, Φ 2 (θ) := F 0 (θ) + θF 2 θM ⊺ 0 Z ⋆ -Z ≺ 0 (12)
hold for all θ ∈ {T min , T max } where

F 0 (θ) = F 00 + θ Sym M 1 M 3 ⊺ P M 0 0 +F 01 -F 02 , F 00 = M ⊺ ∆ S 1 M ∆ + Sym[(M ⊺ ∆ U 1 -M ⊺ δ U 2 )M T ] -M ⊺ δ S 2 M δ F 01 = M ⊺ 2 Φ 0 M 2 -M ⊺ T P M T + M ⊺ 3 QM 3 -M ⊺ 4 QM 4 + h 2 M ⊺ δ RM δ -M ⊺ h RM h , F 02 = (M 12 + M 13 ) ⊺ Z(M 12 + M 13 ), F 1 = Sym[M ⊺ 0 (S 1 M ∆ + U 1 M T )] -M ⊺ T XM T , F 2 = Sym[M ⊺ 0 (S 2 M δ + U 2 M T )] + M ⊺ T XM T with M 12 = M 1 -M 2 , M 13 = M 1 -M 3 and M 0 = A 0 0 BK , M 1 = I 0 0 0 , M 2 = 0 I 0 0 , M 3 = 0 0 I 0 , M 4 = 0 0 0 I , M ∆ = I -I 0 0 , M δ = 0 I -I 0 , M h = 0 0 I -I , M T = M ⊺ 2 M ⊺ 3 ⊺ . Proof: See Appendix A.
As already mentioned in [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF], [START_REF] Briat | A looped-functional approach for robust stability analysis of linear impulsive systems[END_REF], looped-functionals do not need to be positive definite for all τ ∈ [0, T k ], but need instead to satisfy the looping condition [START_REF] Cloosterman | Stability of networked control systems with uncertain time-varying delays[END_REF]. In this regard, the class of looped-functionals is larger than the class of Lyapunov functionals considered, for instance, in [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF], [START_REF] Seuret | Stability analysis for sampled-data systems with a timevarying period[END_REF], [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF], since the looping-condition is a weaker condition than the condition of positive definiteness. Note also that looped-functionals can incorporate information that is usually not considered in Lyapunov functionals. Notably, the term χ k (T k ) is never involved in any of the Lyapunov functionals considered in the literature.

The novelty with respect to previous works is twofold. Firstly, the obtained conditions do not involve any additional decision variables; introduced for instance in the process of deriving the stability conditions from the derivative of the functional. This has been made possible via the use of the bound stated in Lemma 3.1. Secondly, the proposed method allows for the merging of a discrete-time stability criterion (although stated in continuous-time through the use of looped-functionals) to account for aperiodic sampling, together with a discrete-time Lyapunov-Krasovskii functional for dealing with the delay, in a simultaneous way. Until now, this combination of functionals has never been proposed anywhere else in the literature.

B. Robust stability analysis

One of the main advantages of the proposed method based on looped-functionals relies on the possibility for extending the stability conditions to uncertain systems. To this aim, let us consider now that the matrices of the system (3) are uncertain, possibly time-varying, and satisfy:

A(t) B(t) = N i=1 λ i (t) A i B i ( 13
)
where N is a positive integer and the matrices A i and B i , i = 1, . . . , N , are known and of appropriate dimension. The time-varying parameters λ i (t), i = 1, . . . , N , evolve within the N -unit simplex, that is we have

N i=1 λ i (t) = 1, λ i (t) ≥ 0, i = 1, . . . , N, t ≥ 0.
We then have the following result: Corollary 4.2: The uncertain sampled-data system ( 7)-( 13) with delay h ∈ N is asymptotically stable for all

T k := t k+1 -t k ∈ [T min , T max ], k ∈ N if there exist matrices R, Q, Z ∈ S n + , P, X ∈ S 2n , S 1 , S 2 ∈ S n , U 1 , U 2 ∈ R n×n such that the LMIs I I P I I ≻ 0, F i 0 (θ) + θF i 1 θM i 0 ⊺ Z ⋆ -Z ≺ 0, F i 0 (θ) + θF i 2 θM i 0 ⊺ Z ⋆ -Z ≺ 0 (14)
hold for all θ ∈ {T min , T max } with

F i 0 (θ) = F 00 + θ Sym M 1 M 3 ⊺ P M i 0 0 +F 01 -F 02 , F i 1 = Sym[M i 0 ⊺ (S 1 M ∆ + U 1 M T )] -M ⊺ T XM T , F i 2 = Sym[M i 0 ⊺ (S 2 M δ + U 2 M T )] + M ⊺ T XM T , where M i 0 = A i 0 0 B i K
and all the others M 's matrices are defined in Theorem 4.1.

Proof:

The proof follows from the fact that the LMI conditions of Theorem 4.1 are affine in the matrix M 0 . Noting finally that

M 0 (t) = A(t) 0 0 B(t)K = N i=1 λ i (t) A i 0 0 B i K
then the result follows from a simple convexity argument.

V. EXAMPLE

Let us consider the sampled-data system (3) with matrices [START_REF] Zhang | Stability of Networked Control Systems[END_REF]:

A 0 = 0 1 0 -0.1 , B 0 = 0 -0.1
and K = 3.75, 11.5 .

(15) In the case of constant sampling-period T , we can determine, for any delay value h, the maximal sampling-period that still preserves the stability of the system (3)-( 15) by checking at the location of the eigenvalues of the matrix governing the corresponding discrete-time system. These values are upperbounds to the maximal admissible T max in the aperiodic-case since stability regions of aperiodic sampled-data systems are included in those of periodic sampled-data systems. We then use Theorem 4.1 to analyze the system (3)-( 15) in both the periodic and aperiodic sampling cases. The results are shown in Table I and Table II, respectively.

We can see that, in the periodic case, i.e. Table I, the conditions stated in Theorem 4.1 are almost nonconservative when the delay h is small, but that this conservatism increases as the delay h increases. Note, however, that the proposed approach yields results that are competitive with previously obtained ones.

On the other hand, we can see, from Table II, that the method proposed in [START_REF] Seuret | Stability analysis of uncertain sampled-data systems with incremental delay using looped-functionals[END_REF] is able to yield more accurate results than the proposed one in the aperiodic sampling case. This stems from the fact that a tighter bound than the one stated in Lemma 3.1 is used in [START_REF] Seuret | Stability analysis of uncertain sampled-data systems with incremental delay using looped-functionals[END_REF] in order to derive stability criteria. Note, however, that the computational complexity of the current approach scales much better than the approach considered in [START_REF] Seuret | Stability analysis of uncertain sampled-data systems with incremental delay using looped-functionals[END_REF]. The complexity of the conditions derived in this latter reference indeed increases exponentially with the delay h, making the method quite impractical for large values of the delay. In this regard, the proposed approach shows a good tradeoff between accuracy and complexity.

VI. CONCLUSION

A novel way for analyzing stability of periodic and aperiodic uncertain sampled-data systems with discrete-time delays has been presented. Instead of using a discretetime criterion that would prevent the generalization of the approach to uncertain systems with time-varying uncertainties, an alternative approach based on looped-functionals and Lyapunov-Krasovskii functionals has been preferred. The novelty of the method is twofold. First, the stability conditions are free of slack variables, hence numerically less complex than those of the literature, while still being competitive with them in terms of conservatism, at least, for the considered example. Second, the obtained stability conditions are derived from the combination of looped-functionals and Lyapunov-Krasovskii functionals, an approach that has never been explored before.

Future works will be devoted to the extension to the timevarying delay case and on the use of tighter inequalities in order to reduce the overall conservatism of the approach.

APPENDIX

A. Proof of Theorem 4.1

Let us consider the Lyapunov function for the discretetime system (4) given by

V (χ k (τ ), χ h k ) = χ k (τ ) χ k (0) ⊺ P χ k (τ ) χ k (0) + k-1 i=k-h χ ⊺ i (0)Qχ i (0) +h -1 i=-h k-1 j=k+i δ ⊺ i (0)Rδ i (0)
where δ i (0) = χ i+1 (0) -χ i (0). Let us also define the functional V as

T k V(τ, χ k , T k ) = τ (T k -τ ) χ k (T k ) χ k (0) ⊺ X χ k (T k ) χ k (0) +τ (χ k (τ ) -χ k (T k )) ⊺ S 1 (χ k (τ ) -χ k (T k )) +(T k -τ )(χ k (τ ) -χ k (0)) ⊺ S 2 (χ k (τ ) -χ k (0)) +2τ (χ k (τ ) -χ k (T k )) ⊺ U 1 χ k (T k ) χ k (0) +2(T k -τ )(χ k (τ ) -χ k (0)) ⊺ U 2 χ k (T k ) χ k (0) +T 2 k τ 0 χk (s) ⊺ Z χk (s)ds -T k τ T k 0 χk (s) ⊺ Z χk (s)ds (16)
where Z ∈ S n + , S 1 , S 2 ∈ S n , X ∈ S 2n and U 1 , U 2 ∈ R n×2n . Note that this functional verifies the equality

V(0, χ k , T k ) = V(T k , χ k , T k ) = 0 for all T k ∈ [T min , T max ]. Following Theorem 3.3, we consider Ẇk (τ, χ k , χ h k ) = 1 T k Λ k + T k V (χ k (τ ), χ h k ) +T k V(τ, χ k , T k )
where Λ k is defined in Theorem 3.3. The asymptotic stability of the system ( 7) is established if we can prove 1) the positive definiteness of V (χ k (0), χ h k ) and 2) the negative definiteness of Ẇk . Note first that V (χ k (0), χ h k ) is positive definite if Φ 0 , Q and R are also positive definite. This proves the first condition. Let us now focus on the second one and, to this aim, we define the augmented vector

ξ k (τ ) :=     χ k (τ ) χ k (T k ) χ k (0) χ k-h (0)     .
We then have

Λ k = χ k+1 (0) χ k+1 (0) ⊺ P χ k+1 (0) χ k+1 (0) + χ ⊺ k (0)Qχ k (0) - χ k (T k ) χ k (0) ⊺ P χ k (T k ) χ k (0) -χ ⊺ k-h (0)Qχ k-h (0) +h k i=k-h+1 δ ⊺ i (0)Rδ i (0).
Since χ k+1 (0) = χ k (T k ), the previous expression can easily be reformulated in terms of the augmented vector ξ k (τ ).

Applying then Jensen's inequality to the summation term yields

Λ k ≤ ξ ⊺ k (τ ) M ⊺ 2 Φ 0 M 2 - M 2 M 3 ⊺ P M 2 M 3 +M ⊺ 3 QM 3 -M ⊺ 4 QM 4 + h 2 M ⊺ δ RM δ -M ⊺ h RM h ξ k (τ ) = ξ ⊺ k (τ )F 01 ξ k (τ ). (17
) Note that we also have

V (χ k (τ ), χ h k ) = 2 χ k (τ ) χ k (0) ⊺ P χk (τ ) 0 = 2ξ ⊺ k (τ ) M 1 M 3 ⊺ P M 0 0 ξ k (τ ), (18)
and, finally,

T k V(τ, χ k , T k ) = (T k -2τ ) χ k (T k ) χ k (0) ⊺ X χ k (T k ) χ k (0) +(χ k (τ ) -χ k (T k )) ⊺ S 1 (χ k (τ ) -χ k (T k )) -(χ k (τ ) -χ k (0)) ⊺ S 2 (χ k (τ ) -χ k (0)) +2τ χ⊺ k (τ )S 1 (χ k (τ ) -χ k (T k )) +2(T k -τ ) χ⊺ k (τ )S 2 (χ k (τ ) -χ k (0)) +2(χ k (τ ) -χ k (T k )) ⊺ U 1 χ k (T k ) χ k (0) -2(χ k (τ ) -χ k (0)) ⊺ U 2 χ k (T k ) χ k (0) +2 χ⊺ k (τ ) (τ U 1 + (T k -τ )U 2 ) χ k (T k ) χ k (0) +T 2 k χk (τ ) ⊺ Z χk (τ ) -T k T k 0 χk (s) ⊺ Z χk (s)ds. (19)
The previous expression can be rewritten as [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF] where the matrices F 00 , F 1 , F 2 are defined in Theorem 4.1. Splitting then the integral into two parts and applying Jensen's inequality yields

T k V(χ k , τ ) = ξ ⊺ k (τ ) [F 00 + τ F 1 + (T k -τ )F 2 +T 2 k M ⊺ 0 ZM 0 ξ k (τ ) -T k T k 0 χk (s) ⊺ Z χk (s)ds
T k V(χ k , τ ) ≤ ξ ⊺ k (τ ) F 00 + τ F 1 + (T k -τ )F 2 +T 2 k M ⊺ 0 ZM 0 - -M 12 M 13 ⊺ T k T k -τ Z 0 0 T k τ Z -M 12 M 13 ξ k (τ ). ( 21
)
where the matrices F 00 , F 1 , F 2 are defined in Theorem 4.1. By applying then Lemma 3.1 with α = 1 -τ /T k , we then obtain

M ⊺ T k T k -τ Z 0 0
T k V(χ k , τ ) ≤ ξ ⊺ k (τ ) [F 00 -F 02 + τ F 1 + (T k -τ )F 2 + T 2 k M ⊺ 0 ZM 0 ξ k (τ ). (22)
By finally combining [START_REF] Ariba | Simple conditions for L 2 stability and stabilization of networked control systems[END_REF], [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF] and ( 22) together, we obtain that

Ẇk ≤ 1 T k ξ k (τ ) ⊺ [F 0 (T k ) + τ F 1 + (T k -τ )F 2 +T 2 k M ⊺ 0 ZM 0 ) ξ k (τ ). (23)
A Schur complement yields that the (3) is asymptotically stable if the LMI

F 0 (T k ) + τ F 1 + (T k -τ )F 2 T k M ⊺ 0 Z * -Z ≺ 0 (24)
is negative definite for all τ ∈ [0, T k ] and all T k ∈ [T min , T max ]. Since the above LMI is affine in τ (hence convex), to check its negative definiteness over the entire interval [0, T k ], it is necessary and sufficient to check it at the vertices of the set, that is only over the finite set τ ∈ {0, T k }, which yields the two conditions Φ 1 (T k ) ≺ 0 and Φ 2 (T k ) ≺ 0. The end of the proof is obtained by noting that the LMIs Φ 1 (T k ) ≺ 0 and Φ 2 (T k ) ≺ 0 are also affine in T k and the same procedure applies with the set [T min , T max ], finally yielding the conditions [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-andhold devices[END_REF]. The proof is complete.

13 ⊺

 13 . Using the above bound in (21) yields

TABLE I MAXIMAL

 I STABILITY-PRESERVING PERIODIC SAMPLING PERIOD T = Tmax = T min FOR THE SAMPLED-DATA SYSTEM (3)-(15) FOR DIFFERENT VALUES OF THE DELAY h.

	h	0	1	2	5	10
	Theoretical bounds	1.729	0.763	0.463	0.216	0.112
	[27] (with τ = hT )	1.278	0.499	0.333	0.166	0.090
	[33] (with τ = hT )	1.638	0.573	0.371	0.179	0.096
	[23] (with τ = hT )	1.721	0.701	0.431	0.197	0.103
	[25]	1.729	0.763	0.463	- * 1	- * 1
	Theorem 4.1	1.728	0.761	0.448	0.199	0.103
		TABLE II
	MAXIMAL STABILITY-PRESERVING Tmax (WITH T min = 10 -2 ) FOR
	THE APERIODIC SAMPLED-DATA SYSTEM (3)-(15) FOR DIFFERENT
	VALUES OF THE DELAY h.
	h	0	1	2	5	10
	[25]	1.729	0.763 0.463	- * 1	- * 1
	Theorem 4.1	1.708	0.618 0.377	0.176	0.094

The entry "- * " in the tables means that the conditions were not checked due to an unreasonable computational complexity.

This work is supported in part by the ANR project LimICoS contract number 12 BS03 005 01.


A Jezierska

J.-C Pesquet

H Talbot

C Chaux

ITERATIVE POISSON-GAUSSIAN NOISE PARAMETRIC ESTIMATION FOR BLIND IMAGE DENOISING

Keywords: Poisson-Gaussian noise, parameter estimation, Expectation-Maximization, proximal algorithms, discrete optimization

This paper deals with noise parameter estimation from a single image under Poisson-Gaussian noise statistics. The problem is formulated within a mixed discrete-continuous optimization framework. The proposed approach jointly estimates the signal of interest and the noise parameters. This is achieved by introducing an adjustable regularization term inside an optimized criterion, together with a data fidelity error measure. The optimal solution is sought iteratively by alternating the minimization of a label field and of a noise parameter vector. Noise parameters are updated at each iteration using an Expectation-Maximization approach. The proposed algorithm is inspired from a spatial regularization approach for vector quantization. We illustrate the usefulness of our approach on macroconfocal images. The identified noise parameters are applied to a denoising algorithm, so yielding a fully automatic denoising scheme.

INTRODUCTION

Estimation of noise parameters is a classical problem in signal and image processing. For instance, Donoho [START_REF] Donoho | Denoising by soft-thresholding[END_REF] proposed to employ a robust Gaussian variance estimator from the wavelet coefficients of an image, known as the median absolute deviation estimator (MAD). Since then, many alternative methods addressing this problem have been proposed [START_REF] De Stefano | Training methods for image noise level estimation on wavelet components[END_REF], [START_REF] Liu | A fast noise variance estimation algorithm[END_REF], [START_REF] Abramov | Robust local scale estimators for blind evaluation of noise variance in images[END_REF], possibly from a Bayesian perspective, e.g. in [START_REF] Dobigeon | Hierarchical Bayesian sparse image reconstruction with application to MRFM[END_REF]. A more general problem was investigated in [START_REF] Grou-Szabo | A dominant-noise discrimination system for images corrupted by content-independent noises without a priori references[END_REF] where the authors consider the mixture of a white Gaussian noise and a random impulsive noise.

Unfortunately, in many cases, the imaging noise is not simply additive. Due to the quantum nature of light, the images are degraded by shot noise. Thermal noise may also play a prominent role in some modalities, e.g. astronomical imaging. Although a Gaussian approximation is sufficient in some cases, one may need to consider a noise model where the mean or the variance depends on the image intensity. Consequently, studies related to image recovery in the presence of signal dependent noise constitutes an active research area. Examples include restoration methods in the presence of Poisson noise [START_REF] Pustelnik | Parallel proximal algorithm for image restoration using hybrid regularization[END_REF] and more recently Poisson-Gaussian models [START_REF] Luisier | Image denoising in mixed PoissonGaussian noise[END_REF], [START_REF] Jezierska | A primal-dual proximal splitting approach for restoring data corrupted with Poisson-Gaussian noise[END_REF]. However, the associated noise identification problems have not yet been extensively studied in the literature.

In this paper, we address the problem of estimating the parameters of a Poisson-Gaussian noise from a single image. The problem is formulated within a variational framework. The proposed alternating optimization algorithm jointly estimates the signal of interest and the noise parameters from a single noise realization. We consider various choices of regularization functions promoting sparsity, including nonconvex ones.

The estimation of noise parameters in the presence of multiplicative or Poisson noise is more challenging than for additive stationary Gaussian noise. The additional challenge stems from the noise nonstationarity. There have been many attempts to address this issue, for instance in the context of CCD imaging [START_REF] Foi | Practical Poissonian-Gaussian noise modeling and fitting for singleimage raw-data[END_REF], radar [START_REF] Lukin | Filtering of radar images based on blind evaluation of noise characteristics[END_REF], and confocal microscopy [START_REF] Paul | Automatic noise quantification for confocal fluorescence microscopy images[END_REF]. However, all these approaches share a similar structure. Namely, they proceed by first determining pixel groups with uniform intensities [START_REF] Abramov | Improved method for blind estimation of the variance of mixed noise using weighted LMS line fitting algorithm[END_REF], [START_REF] Liao | Noise estimation for single-slice sinogram of low-dose x-ray computed tomography using homogenous patch[END_REF] followed by a parameter fitting procedure. Similarly to most of these techniques, our approach includes both a segmentation and an estimation process. However, unlike the previous works, the two main steps are repeated iteratively. At each iteration, the new partition and noise parameter estimates are obtained by minimizing a cost function.

The remaining of this paper is organized as follows. The model is introduced in Section 2. Then, the considered parameter estimation problem and the proposed methodological and numerical solutions are described in Section 3. We provide experimental results in Section 4, where the performance of our approach is illustrated using macroconfocal data. Finally, Section 5 concludes the paper.

MODEL

Let x be the original image of size N × M . Its values are assumed to belong to a finite set {u1, . . . , uK } where K ∈ N * is the number of distinct intensity values. We will denote by u the vector [u1, . . . , uK ] ⊤ ∈ R K . We consider a partition P = (D k ) 1≤k≤K of the image support S = {1, . . . , N } × {1, . . . , M }, which is related to the image x through the following relation:

x = (u i P (s) ) s∈S ∈ {u1, . . . , uK } N ×M , (1)
where i P (s) s∈S ∈ I = {1, . . . , K} N ×M is a label image defined as

(∀s ∈ S)(∀k ∈ {1, . . . , K}) i P (s) = k ⇔ s ∈ D k . ( 2
)
Since x is a function of i P and u, it will be denoted by xi P ,u throughout this paper. The observed noisy image y ∈ R N ×M is a realization of a random vector Y and is such that

y = α q + w (3)
where α is a positive scaling factor, and q = (q(s)) s∈S (resp. w) are realizations of mutually independent random vectors Q (resp. W = (W (s)) s∈S ) with independent components. More precisely,

Q ∼ P(xi P ,u) (4)
(∀s ∈ S) W (s) ∼ N (c, σ 2 ), (5)
where c ∈ R and σ ∈ (0, +∞) are the mean and the standard deviation of the Gaussian noise component, respectively, and P(x) denotes a multivariate Poisson distribution with mean vector x.

Then, the parameter estimation problem corresponds to finding a vector of unknown noise parameters θ ∈ T , where θ = [c, σ 2 , α, u ⊤ ] ⊤ and T is the parameter set which is here equal to R × (0, +∞) 2 × C, C being a closed convex subset of R K . The unknown vector θ is related to the vector of observations y through the probability density function of Y , which has the following form: for every y = y(s

) s∈S ∈ R N ×M , pY (y; i P , θ) = K k=1 s∈D k   +∞ q(s)=0 e -u k (u k ) q(s) q(s)! e - (y(s)-αq(s)-c) 2 2σ 2 √ 2πσ 2   . (6)

PROPOSED APPROACH

Problem formulation

We adopt a variational approach where the parameter vector θ is estimated by minimizing a penalized criterion. We then solve minimize

(i P ,θ)∈I×T Φ(θ, i P , y) + ρ(i P ). (7)
The above function Φ is defined as the neg-log-likelihood of y:

Φ(θ, i P , y) = -log pY (y; i P , θ) (8)
and ρ is a regularization function aiming at incorporating a priori information about the homogeneity of the level sets of the image. Note that Problem ( 7) is nonconvex due to the fact that i P belongs to a (nonconvex) set of discrete values.

Algorithm

We propose the following alternating optimization algorithm:

Algorithm 1 Proposed algorithm Initialization: Fix K ∈ N * and θ (0) ∈ T . Main loop: For ℓ = 0, 1, . . .      i (ℓ) P ∈ argmin i P ∈I Φ(θ (ℓ) , i P , y) + ρ(i P ) θ (ℓ+1) ∈ argmin θ∈T Φ(θ, i (ℓ)
P , y)

Assuming that noise parameters (c, σ 2 , α) are known, the above algorithm reduces to the joint denoising-quantization algorithm proposed in [START_REF] Chaux | A spatial regularization approach for vector quantization[END_REF]. In particular, the first step of Algorithm 1, which consists of searching i P , is unchanged. Classical discrete optimization techniques are used for this purpose [START_REF] Murota | Algorithms in discrete convex analysis[END_REF]. Additional difficulties arise in the second step of Algorithm 1 since the minimized function is no longer convex. However, we are able to use some of the results of [START_REF] Jezierska | An EM approach for time-variant Poisson-Gaussian model parameter estimation[END_REF], where the problem of estimating Poisson-Gaussian noise parameters from time series is addressed. The proposed method however required a large enough number of image realizations in order to provide consistent estimates of the parameters.

Parameter update

In order to solve numerically the nonconvex problem stated in the second step of Algorithm 1, we propose to employ the Expectation-Maximization (EM) algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF]. In such settings, the EM algorithm generates a sequence of estimates (θ (ℓ) ) ℓ∈N which is given by

(∀ℓ ∈ N) θ (ℓ+1) = argmin θ∈T Φ(θ | θ (ℓ) ) (9)
where

Φ(θ | θ (ℓ) ) = 1 2σ 2 s∈S E Q|Y =y,θ (ℓ) [(y(s) -αQ(s) -c) 2 ] + N M 2 ln(σ 2 ) + K k=1 card(D (ℓ) k )u k - K k=1 ln u k s∈D (ℓ) k E Q|R=r,θ (ℓ) [Q(s)] (10)
and (Q(s)) s∈S are the components of Q as defined in (4) with i P = i

P . This leads to the following operations to be performed at each iteration ℓ:

1. Find u (ℓ+1) as a solution of

u (ℓ+1) ∈ argmin u∈R K ϑ(u) (11)
where ϑ(u) = K k=1 ϑ k (u k ), and, for every k ∈ {1, . . . , K},

ϑ k (u k ) = card(D (ℓ) k )u k -ln u k s∈D (ℓ) k E Q|R=r,θ (ℓ) [Q(s)]. (12)
In the unconstrained case when C = R K , we have then

u (ℓ+1) = s∈D (ℓ) k E Q|R=r,θ (ℓ) [Q(s)] card(D (ℓ) k ) . ( 13
)
On the other hand, for some regularization terms [START_REF] Chaux | A spatial regularization approach for vector quantization[END_REF], set C is useful to impose a total order constraint. This amounts to minimizing ϑ under the constraints that

V u ∈ D = [δ, +∞[×[0, +∞[ K-1
where δ is some small positive value, and V is the linear operator defined as

V : R K → R K (u1, . . . , uK ) → u1, u2 -u1, . . . , uK -uK-1 . ( 14
)
2. Determine c (ℓ+1) and α (ℓ+1) by solving the following system of linear equations:

   N M s E Q|Y =y,θ (ℓ) [Q(s)] s E Q|Y =y,θ (ℓ) [Q(s)] s E Q|Y =y,θ (ℓ) [Q(s) 2 ]    c (ℓ+1) α (ℓ+1) =    s y(s) s y(s)E Q|Y =y,θ (ℓ) [Q(s)]    . (15) 3. Compute (σ 2 ) (ℓ+1) as s y(s) y(s) -α (ℓ+1) E Q|Y =y,θ (ℓ) [Q(s)] -c (ℓ+1) N M . ( 16
)
Note that even when a closed form solution to Problem [START_REF] Lukin | Filtering of radar images based on blind evaluation of noise characteristics[END_REF] does not exist, it can still be solved efficiently using proximal tools. We propose to solve numerically this problem by using a primal-dual proximal algorithm [START_REF] Combettes | Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators[END_REF].

Algorithm 2 Primal-dual algorithm for solving [START_REF] Lukin | Filtering of radar images based on blind evaluation of noise characteristics[END_REF] Initialization γ ∈ (0, +∞), u

1 ∈ R K , v (0)
∈ R K . Main loop For j = 0, . . .            w (j) 1 = u (j) -γV ⊤ v (j) 1 p (j) 1 = prox γϑ (w (j) 1 ) w (j) 2 = v (j) 1 + γV u (j) p (j) 2 = w (j) 2 -γPD(γ -1 w (j) 2 ) v (j+1) 1 = v (j) 1 -w (j) 2 + p (j) 2 + γV p (j) 1 u (j+1) = u (j) -w (j) 1 + p (j) 1 -γV ⊤ p (j) 2 (0) 1
In the above algorithm, we denote the projector onto the closed convex set D by PD. The proximity operator of γϑ with γ > 0 admits a closed form expression [START_REF] Combettes | Proximal splitting methods in signal processing[END_REF] and it is given by [START_REF] Jezierska | An EM approach for time-variant Poisson-Gaussian model parameter estimation[END_REF] where, for every k ∈ {1, . . . , K},

∀u = (u k ) 1≤k≤K ∈ R K prox γϑ (u) = prox γϑ k (u k ) 1≤k≤K
prox γϑ k (u k ) = 1 2 u k -card(D k ) (18)
+ |u k -card(D k )| 2 + 4 s∈D k E Q|R=r,θ (ℓ) [Q(s)] .
The convergence of the primal-dual algorithm is guaranteed, using the following result deduced from [START_REF] Combettes | Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators[END_REF]Theorem 4.2].

Theorem 3.1 Under the assumptions that γ ∈ [ǫ, (1ǫ)/β] where ǫ ∈ (0, 1/(β + 1)) and β = V , there exists a minimizer u of [START_REF] Lukin | Filtering of radar images based on blind evaluation of noise characteristics[END_REF] such that the sequences u (j) j∈N and p

(j) 1 j∈N converge to u.
Concerning the computation of the required conditional expectation values, it can be shown that, for every s ∈ S,

E Q|Y =y,θ (ℓ) [Q(s)] = ζ (ℓ) s η (ℓ) s and E Q|Y =y,θ (ℓ) [Q(s) 2 ] = ξ (ℓ) s η (ℓ) s (19)
where

ζs(θ (ℓ) ) = ∞ q(s)=0
Πs(θ (ℓ) , 1, q(s)) ( 20)

ηs(θ (ℓ) ) = ∞ q(s)=0
Πs(θ (ℓ) , 0, q(s)) ( 21)

ξs(θ (ℓ) ) = ∞ q(s)=0
Πs(θ (ℓ) , 1, q(s))

+ ∞ q(s)=0
Πs(θ (ℓ) , 2, q(s))

In these formulas, for every (d, q(s))

∈ N 2 , Πs(θ, d, q(s)) = exp - (y(s) -α(q(s) + d) -c) 2 2σ 2 × (u i (ℓ) P (s) ) q(s)+d q(s)! . (23)
The infinite summations in ( 20), ( 21) and ( 22) are approximated by finite sums with the following bounds:

q(s) + = q(s) * + ∆ σ α , q(s) -= q(s) * -∆ σ α
where q(s) * = σ 2 α 2 W α 2 σ 2 u i (ℓ) P (s) e α σ 2 (y(s)-c-dα) . Here, W denotes the Lambert function and ∆ > 0. This choice has been shown to ensure a fast decay of the approximation error as a function of ∆.

Patch-based initialization

Since Problem ( 7) is nonconvex, the proposed approach is sensitive to initialization. In the following, we propose an initialization procedure based on image patches [START_REF] Salmon | Poisson noise reduction with Non-Local PCA[END_REF]. Firstly, the observed image y is decomposed into non overlapping patches Θy = {Θy1, . . . , ΘyO}, where O stands for the total number of patches. Next, we estimate the vectors mean (Θy) ∈ R O and Var (Θy) ∈ R O , whose elements are the mean and the variance over the pixels belonging to each patch, respectively. Under the assumption that the intensity of each patch is constant, the mean and variance of the patch is given by: for every o ∈ {1, . . . , O},

[mean (Θy)] o = αυo + c (24) [var (Θy)] o = α 2 υo + σ 2 (25)
where υo ∈ {u1, . . . , uK }. Although, the assumption about the constant intensity of each patch is not exactly satisfied, an approximation θ (0) of the parameter vector is given by

• α (0) = [Var (Θy)] o 1 / [mean (Θy)] o 1 , where o1 = argmax o∈{1,...,O} [mean (Θy)] o • c (0) = min o∈{1,...,O} [mean (Θy)] o • σ 2 (0) = [Var (Θy)] o 2 ,
where o2 = argmin o∈{1,...,O} [mean (Θy)] o .

• u (0) 1 = ǫ, where ǫ is a small value greater than 0, and (∀k ∈ {2, . . . , K}) u

(0) k = u (0)
k-1 + umax/(K -1). The maximum intensity value umax of x is assumed to be known.

Note that in contrast with the approach proposed in [START_REF] Jezierska | An EM approach for time-variant Poisson-Gaussian model parameter estimation[END_REF] the EM algorithm is initialized with an improved set of parameters at each iteration.

RESULTS

We evaluate the robustness of our model under different working conditions. For all the presented experiments, the patch size is 12 × 12, K = 20, and ǫ = 0.2. The regularization ρ is an anisotropic total variation-like function defined as follows: let (n, m) denote the coordinates of a generic pixel s, We set the regularization parameter µ to 1.3 when ψ is set to a binary cost function defined as

ρ(i P ) = µ N -1
(∀ξ ∈ [0, +∞)) ψ(ξ) = 0 if ξ = 0 1 otherwise, (27)
and 0.8 when ψ is the identity function.

The results of the time series noise identification procedure described in [START_REF] Jezierska | An EM approach for time-variant Poisson-Gaussian model parameter estimation[END_REF] provide a high quality image, akin to a ground truth for real microscopy data (Figs 1 (a,e)). In order to show the versatility of our model, we use ψ defined as the identity function for the experiments presented in Figs 1 (a-d), while the binary cost function ( 27) is used for the experiment illustrated in Figs 1 (e-h). For the identity function, a total order constraint needs to be enforced in the EM step. For these two experiments, umax is set to 150 and 40 respectively. In the first case, the image in Fig. 1 In Fig. 1, the estimated noise parameters are subject to small remaining errors with respect to the true values. The quantized image, a side result of our procedure, is also displayed. These results show that our procedure provides reliable results on real data that can be fed to a denoising procedure, e.g. [START_REF] Luisier | Image denoising in mixed PoissonGaussian noise[END_REF]. The corresponding PSNR values are 28.45 dB and 28.82 dB for the images in Figs 1(d) and 1(h), respectively. For comparison, and to emphasize the importance of accurate parameter estimation, we averaged the PSNR of 50 image denoising realizations. These had parameters set around the optimal value, i.e we allowed parameters to vary following a Gaussian distribution with a standard deviation of 10% of the optimal value. This resulted in PSNRs of only 26.43 dB and 26.20 dB, respectively.

CONCLUSIONS

We have proposed a reliable Poisson-Gaussian noise parameter estimation method for imaging applications, which only requires a single noisy image as input. The derived alternating minimization procedure allows us to estimate the noise parameters required by many denoising procedures. Here we used the method proposed in [START_REF] Luisier | Image denoising in mixed PoissonGaussian noise[END_REF]. However, other denoising procedures can be used as well. The simulation results demonstrate the usefulness of our approach on confocal images. To our knowledge, this is the first blind estimation method for handling Poisson-Gaussian noise with nonzero background. In our future work, we plan to extend the proposed image noise estimation approach and to investigate its use for other parametric forms of complex noise probability distributions.

  P (n + 1, m)i P (n, m)|) P (n, m + 1)i P (n, m)|) , µ ≥ 0. (26)

  (a) was corrupted by Poisson-Gaussian noise with parameters α = 5, c = 150, and σ 2 = 125. The identified noise parameters were α = 4.89, c = 156.35, and σ 2 = 115.2 (Fig. 1 (c)). In the second case of Fig. 1 (e) the estimated value of α = 20, c = 150, and σ 2 = 1000 were α = 19.5, c = 142, and σ 2 = 907 (Fig. 1 (g)).

Fig. 1 .

 1 Fig. 1. Single image noise identification results.

Acknowledgement We thank Florian Luisier for providing us with the PURE-LET code.

This work was supported by the Agence Nationale de la Recherche under grant ANR-09-EMER-004-03.


Jean-Philippe Béja

FORBIDDEN MEMORY, UNWRITTEN HISTORY THE DIFFICULTY TO STRUCTURE AN OPPOSITION MOVEMENT IN THE PRC

The paper tries to show how the control over the transmission of memory by the Party has hindered the structuring of an opposition movement. It analyses the cases of the anti-rightist movement and of the 1989 pro-democracy demonstrations to illustrate this thesis. Since its foundation in 1921 the Chinese Communist Party has attached great importance to the writing of history. It has also been keen on establishing its control over memory. Ritualised and manipulated in the "accounts of bitterness", memory has represented an important tool in the repertoire of CCP propaganda, and has played a considerable role in the education of the youths. On the other hand, memory of past episodes of resistance to CCP rule has been strictly obliterated. The anti-rightist campaign is a good example of the CCP policy toward memory. Control of discourse in the private sphere was such that even the heirs of the 1957 protesters were unaware of the contents of their protest. The members of the Red Guard generation accepted the official discourse which presented them as "enemies of the people". Therefore, they could not build on the analysis they had produced and had only a very limited set of concepts at their disposal to understand the nature of the regime. Interruption in the process of memory was therefore a hindrance to the development of opposition forces. It took the Red Guard/rusticated youth generation more than ten years to realise that the discourse of the rightists could become part of the political heritage necessary to build a resistance movement Limitations put by the Party on transmission of memory was also instrumental in the relatively low level of political culture of the students during the 1989 pro-democracy movement. Their ignorance of the theoretical debates which had taken place during the Democracy Wall (1978-79) kept them from building upon an already existing body of texts criticising party rule. Every resistance movement since 1949 has had to start from scratch. The difficult transmission of memory has prevented the accumulation of experience by the pro-democracy movement. Every episode of resistance appears isolated to actors who view themselves as innovators. The 1989 students were convinced that they had been the first to have launched a real challenge to the Party in the history of the PRC, and did not view themselves as the successors of a long line of resisters. This inability to capitalise on the memory of previous episodes of resistance, which is due to the Party's ban on remembrance and to its monopoly over interpretation of history, has resulted in a segmentation which has had a very negative impact on the structuring of a political opposition.

FORBIDDEN MEMORY, UNWRITTEN HISTORY THE DIFFICULTY TO STRUCTURE AN OPPOSITION MOVEMENT IN THE PRC Jean-Philippe Béja

Senior Researcher, CNRS/CERI, Paris New Chinese dynasties have always legitimized their power through historical justifications. After they had consolidated their rules, one of their first acts consisted in writing the history of the preceding dynasty. Although this tradition has been interrupted by the establishment of the Republic in 1911, it has remained very important for Chinese rulers to ground their power in history.

This tendency has been reinforced by the fact that since 1949, China has been ruled by a Communist party. Ever since the 1917 October Revolution, Communist regimes have been famous for their ability to manipulate history. Comparing photographs of the leaders in different periods and seeing who had been erased has been a pastime of many a Sovietologists.

The Chinese communist party has inherited both these traditions, and mobilization of the past in order to legitimate its rule has been all pervasive.

Anniversaries of various revolutionary episodes punctuate the lives of Chinese citizens.

The founding of the Communist Party on July 1 st , Army Day on August 1 st , the anniversary of the May 4 th movement, the proclamation of the People's Republic on October 1 st provide opportunities to celebrate the unity of the people behind its avant-garde, and -since television has made its way into every citizen's home-and to eulogize the personalities of old Revolutionaries through interviews. History's presence is manifest even in the social structure itself, as the date of participation in Revolutionary activities (canjia geming) is a criterion for promotion on the bureaucratic scale --the earlier, the higher. These dates are obviously selected according to political criteria, and since the founding of the People's Republic, the list of celebrations has been through many changes.

Whereas some of them pertain to the continuity of the Party rule, like the abovementioned ones, many are linked to a certain political line: whereas July 16 th was celebrated all over China from 1967 to 1976 1 , nobody ever thinks any more of mentioning such a date which brings back bad memories. The importance of historical dates is quite striking and was revealed during the Cultural Revolution when most Red Guards groups were named after the date of a crucial event for their existence.

However, manipulation of history and celebration of anniversaries are not an exclusive characteristic of CCP rule. All political regimes feel the need to ground their legitimacy in history, and democratic regimes are no exception. Besides, official history is always more or less manipulated by the party in power. However, in the case of the People's Republic of China, the difference is that there is only official history, and that freedom of discussion in this area has always been either totally absent or very strictly limited. The following Soviet anecdote could very well apply to the Middle Kingdom. "Armenian Radio is asked:

'Is it possible to foretell the future?' Answer: 'Yes', that is no problem: we know exactly what the future will be like. Our problem is with the past: that keeps changing" 2

The (re)writing of history has been a very important exercise in all socialist countries as their political leaders are supposed to have been designated by the laws of human development in order to accomplish the historical task of leading the people towards the glorious goal of Communism. So, when a leader falls in disgrace, he has to be erased from history books, and can be remembered only in a negative way. The case of Liu Shaoqi is quite revealing. Whereas during the fifties he had been presented as an major leader of the working class movement before Liberation, he was labeled a scab during the Cultural Revolution, and kids were educated to hate his name. The same happened to Marshall Lin Biao whose name was erased from the campaigns of the War of Liberation after he "betrayed" the Great Helmsman in 1971. This is not the place to detail how leaders have been first cleansed, then reinstated in official history. But the numbers of functionaries affected to this task, the care with which everything is periodically re-written show that the monopoly of interpretation of the past is considered as an essential instrument of political control by Communist parties all over the world. During the Cultural Revolution, this mania reached ridiculous extremes: subscribers to the Renmin ribao regularly received notices asking them to scrap a page on a past issue because the author of the article had fallen from grace or because a piece of news had eulogized a person since them labeled a counter-revolutionary. The need to control the past was so strong that during the late sixties and the early seventies, readers in public libraries could not consult an issue of the People's Daily dating back more than one year. Mao, London, Chatto and Windus, 1994, p.406). After that, every July 16 th , all the Chinese citizens who could swim dived in rivers, lakes and in the sea to emulate the Great Leader. 2 Quoted in Rubie Watson (Ed.) Memory, History, and Opposition, Santa Fe, School of American Research Press, 1994, p.1

CONTROL OF MEMORY: AN IMPORTANT ASPECT OF POLITICAL EDUCATION

But controlling the official record of events is not enough. Besides mobilizing history to its service, the Chinese Communist Party has also attached a particular importance to controlling memory. In the days of Mao Zedong, it was an important part of what was then called "class education" (jieji jiaoyu). Whenever sent to the countryside or to a factory, even for a short visit, pupils from primary and secondary schools had to listen to old peasants or old farmers recall the suffering of the past --yiku sitian (recalling the bitter [past] to appreciate the sweet [present])-During the sixties and the seventies, these represented an important part of the political education of the youths.

These were not descriptions of historical events, but stories of ordinary people which were supposed to make young people feel what life in the old society had been like, and realize how lucky they were to live under the glorious leadership of the Communist party.

The story telling was very ritualized. It always started with a description of the utter poverty that reigned at the time (in factories, workers worked long hours for the capitalists and lived in huts; whereas in the countryside, they were used as farm hands without any land, compelled to do the hardest jobs for the landlords), of the hunger which was all pervasive. Then, the story reached a climax when the landlord (or the capitalist) resorted to violence against the story teller (rapes were never clearly mentioned by women, but the audience could imagine it as lust was presented as a typical feature of exploiters). The poor tried to resist but they were impotent until one of them (usually the story teller) got in touch with the Red Army (or the People's Liberation Army) which finally came to liberate the village (or the town) and installed the people's power. In the seventies in some places 3 , the story did not end at this point, because, very quickly, many communists revealed themselves to be rightists, revisionists, agents of the landlords, and ordinary citizens had to wait until the Cultural Revolution was launched "by Chairman Mao himself" to be really liberated. These stories were manipulated by the leadership so as to serve the present struggle. For example, in some places during the seventies, yi ku designated the suffering of ordinary people both under the Guomindang regime and during the "seventeen years of the black revisionist line", and the negative characters were as much the revisionist cadres as the landlords (or capitalists) 4 . In the late seventies, the situation was reversed and the bad ones became the "Gang of Four" and the "ambitious" rebels. These manipulated and ritualized memories have always constituted an important resource in the repertoire of CCP propaganda. Usually compiled under the supervision of Party secretaries, they obviously followed the political line of the moment and the roles were assigned accordingly. The tales of bitterness were considered a very efficient complement to the official version of history taught in schools through manuals because they appealed to the feelings of pupils whereas the manuals appealed to reason. Under maoism, "class feelings" (jieji ganqing) were an important part of the character of the "new person", perhaps all the more so as feelings towards the other sex were forcefully repressed. All the persons who have been educated during the fifties and sixties have been deeply impressed by these stories, which were part of any school outing. The appeal to feelings was so manipulated that during the Cultural Revolution, cadres organised yiku fan ("remembering the bitter" meals) when urban youths (but also peasants) had to eat wild vegetables and all sorts of terrible foods peasants were reduced to consume before Liberation5 . This emphasis on the use of a manipulated personal memory to convey the political message of the moment shows that communist party leaders were aware of the importance of direct personal contact in the psychological formation of the youths. It was actually quite efficient. As Xu Youyu noted about the youths during the Cultural Revolution: "Although two years of political movement had provoked doubt and suspicions towards political faith, certain ideas were deeply rooted in our minds, such as 'the old and the new society are two completely different worlds', 'the complete liberation of poor peasants' and that type of propaganda."6 . These ideas went obviously hand in hand with hatred for landlords and agents of the Old society, but also for peasants who had resisted forced collectivization. In this narrative, the latter, who actually had often supported the Party's struggle before Liberation, but had opposed its policies in the mid-fifties, were considered as identical to the reactionaries. Therefore, opposition to the Communist Party equaled opposition to the people. This aspect of the Party's monopoly over memory is one of the factors explaining that it took a very long time even to the most radical Red Guards to question the legitimacy of the regime itself.

Nevertheless, some examples show that even the most totalitarian regimes are unable to totally control the expression of memory. In a village in Henan, a rusticated youth was moved to tears by a poor and middle peasant's account of famine. In the discussion that ensued (an essential part of the unfolding of this typical seance of mobilization ), the youths gave vent to their hatred of the Guomindang landlords and their feudal allies, whose policies had led their class brothers to famine. To which the peasant replied: "But I am not talking of the days before Liberation. In those times we had plenty to eat.

No, I'm talking about the terrible years 1960 and 61"(the Great Leap Forward)"7 . This is not an isolated case. Let's listen to Xu Youyu, writing about peasants in Sichuan. " Before the meetings, cadres explained many times to old peasants that while recalling the bitter, they should not recall the wrong bitterness, they should recall pre-1949 bitterness, and not the 1962 bitterness. Despite that, old people whose mind was not very clear talked about their sufferings in 1962, describing how terrible the famine had been, how so many people died of hunger. Because of the Cultural Revolution propaganda, they would put the fault on Liu Shaoqi, but angry cadres would shout at them: 'we ask you to remember Chiang Kai-shek's bitterness, not Liu Shaoqi's!"8 . These slips in the working of the propaganda machinery helped the 1960s youths find out that propaganda was full of lies and that allout collectivization was loathed by all peasants, not only by the landlords' offspring as official discourse had it. These uncontrolled memories therefore played an important role in the political formation of many a member of the "rusticated youth" generation and represented an essential step on the road to organizing opposition.

UNWRITTEN HISTORY

Of course, the more importance the Party attaches to the discourse about the past, the more it tries to control it. In a regime where no regular elections come to comfort the legitimacy of the leaders, history is an important asset. But it is rarely as smooth as the Party wants it to be. Often, the official organizations, such as the PLA, the Central committee, have not played the exact role that has been taught in history manuals which tend to present the conquest of power as an heroic struggle by Party members under the leadership of the Central Committee. Therefore, historical events which do not corroborate this thesis cannot even be mentioned, and personal or family memories which differ from the Party line are chased with unbelievable intensity.

The writer Zhang Kangkang, in a novelette entitled Collective Memory 9 shows that even in the twenty first century some events, although accomplished by Communists during the Liberation war, have been expelled from history, and even from memory. The story tells of a Party historian who must write a piece about the Liberation of his city. He finds out that a woman who had studied abroad and become a communist upon her return to China during the anti Japanese war, had played a very important part in this event. Through her love for an engineer working with the KMT, she was able to save from destruction a strategic bridge which the PLA needed in order to continue its advance. When the historian tries to get information about that event, he can't find a record in the archives, and cannot get any confirmation from the old general who has entered history books as the liberator of the city. In his interview with the historian, the latter insists on the necessity to "take the big picture into account" and dwell upon the role of the PLA. Through his investigation, the historian will finally find the woman. However, having been labeled a rightist in 1957, she has hardly been rehabilitated. Still living in a small crammed apartment, she has lost all illusions, and refuses to tell him her story, stating clearly that her personal memory will never be allowed to challenge the official version of written history. The story ends with the article being censored by the historian's superior.

Although this story is presented as fiction, it actually refers to real events. This shows that without the consent of the upper echelons, even a piece of history which has not been deemed negative by the Party, cannot make its way into collective memory, with the result that it obliterates even personal memory.

Naturally, episodes of resistance to Party rule have been eliminated more thoroughly both from official history and from personal memories. "The colonization of public and private space is one of the hallmarks of state socialism"10 .

TO REMEMBER IS TO FORGET

Forgetfulness is an indispensable element in the process of memory. No memory is possible if all the details of a person's life are remembered: memory is a process of selection. The Party's memory therefore resorts to forgetfulness. But far from being a natural process as in the workings of memory as analyzed by philosophers, in this case it is officially decreed and achieved by teams of specialists. According to the political line of the moment, leaders indicate which part of past experience should be remembered --for example, class origin has played a very important role during the first three decades of the people's Republic --and which should be forgotten --the acts of resistance by peasants, by rightists or all kinds of so-called counter-revolutionaries.

During its first decades in power, when it was able to carry the adhesion of the broad masses of the people, the Party was so keen on controlling the minds of citizens, especially the youths, that it made it quasi-impossible for people to transmit their personal ideas and memories to their heirs even inside the family. "If the family head's political attitude coincided with the official one, he would encourage his children to become activists with all his strength. Otherwise, the great majority would adopt an attitude encouraging their sons and daughters to follow the official line, for the sake of their children's future, and in many cases, people were afraid that if they expressed their antiofficial attitude, they could be denounced by their children". And in fact, there were many instances when children "traced the line of demarcation from their parents"(huaqing jiexian) 11 . Besides the fact that when a person became the target of a movement, his (her) spouse divorced him (her) so as not to be considered counter-revolutionary too. Those who refused to do so never talked about the husband (wife) who had been labeled a reactionary , and often sent to the countryside for re-education, if not to jail. They especially refrained from presenting his (her) ideas objectively to their children so that they could judge by themselves. Pressure was such that even inside the warmth of a home, family members resorted to official discourse when talking to each other. People would use newspaper language to refer to the "crimes" of their spouses so that children would not "make mistake" when going to school. Therefore, memory, as the transmission of the way public events --especially episodes of resistance--have been experienced by the people, has been very carefully controlled by the authorities under maoism, so that, for example, even the rightists' heirs were not aware of the contents of their parents' discourse 12 . To them, they were counterrevolutionaries who, volens nolens, had spoken against the people. So when they had the opportunity to vent their frustrations against the regime, when Mao called upon them to "bomb the bourgeois headquarters", they could not use the experience of the previous generation. It would take them many years to cover the distance that had cost their fathers so much.

11 Xu Youyu, "'Wenge' de qiyin" (The root causes of the Cultural Revolution), in Xu Youyu, op.cit., p.141. This is not a specific feature of China: Pavel Korchaghin, the hero of How Steel was tempered, made himself famous by denouncing his "reactionary" parents. In 2000, this book was in a prominent place in many bookstores in Peking. 12 In his autobiography, Wei Jingsheng recalls the story of a rightist he met in Xinjiang during the Cultural Revolution. When he went to tell her husband in the city how she was faring, "he refused to admit that he had any relation with his children's mother". See "Autobiographie de Wei Jingsheng"(Wei Jingsheng's autobiography), in Sidane, V, Zafanolli, W.Procès politiques à Pékin, (Political trials in Péking) Paris, Maspero, 1981, pp.65-66.

THE MEMORIES OF THE RIGHTISTS AND THE RED GUARDS

During the Cultural revolution, and the subsequent rustication of youths, many young urban dwellers who blindly believed in Party propaganda and obeyed Mao's call were convinced that the 1957 rightists had been counter-revolutionaries who opposed the people. They did not hesitate to persecute and beat them in the early years of the movement. Therefore, in their opposition to the excesses of what they termed the "new bourgeoisie", they never used the political resources of the Hundred Flowers movement.

This was out of utter ignorance and because of the obliteration of memory. They started to understand that the rightists' ideas were not so outrageous when they met them either during the "Great exchange of experiences" (da chuanlian), or later when they were sent to the countryside. It is worth quoting at length Wei Jingsheng's experience with a rightist.

In 1966, Wei was a radical Red guard --of "good" class background as his father was a military officer--and during the movement he had been involved with a group (Liandong, the United Action Committee) which had adopted a very violent attitude against most "counter-revolutionaries". But despite his prejudices, reality made him change his mind and helped him set on the road to opposition: " ... an "old rightist" we had met [during our travels in Xinjiang] confessed to us that she would never have thought, when she entered the Party, that one day it could have been controlled by such cruel and inhumane individuals as those who were now in power. Such a discourse shocked us at the beginning, and we thought that she deserved her "rightist" 's label. But ... she told us many concrete stories, including her own, which granted her words an authority that was difficult to deny and aroused a great interest in me...The 'old rightist's' experience also called for the following question: 'Why are good people always defeated and why do bad ones always triumph?' Although she appeared very ill at ease with us because of her rightist background, I thought she was a good person and that people like her would never willingly have hurt the people's interest. To label her a 'Rightist' had been wrong in the first place and it was a scandal that such an honest and competent person had been deported to that far-away village...." 13 Wei is not the only person engaged in resistance whom the ignorance of the past has delayed in his process of awareness. Zhang Musheng was the son of a leading cadre who had gone voluntarily to the countryside in Inner Mongolia in 1965. His stay in this remote place had convinced him that as the peasants were hostile to collectivization, they were "reactionary". But in 1966, thanks to another rusticated youth who had successfully adopted a policy of decollectivization in an isolated brigade he led, he started to wonder whether he had been right. After all, the problem might lie with the Party, not with the people. " So he started to read prohibited books relating the history of the communist international. He also read Chinese 'rightists' books, such as Ma Yanshu's speech denouncing the Party's world (dang tianxia), writings by Chu Anping and big volumes of rightists' opinions and criticism of the Party rule, Ding Ling's "March 8 th is good" and Wang Shiwei's "Wild lilies'....He compared Liu's and Mao's speeches at the 7000 cadres' meeting in 1962 and found out that China's problem could not be solved by applying only class struggle...His personal experience in the countryside led him to completely reevaluate Peng Dehuai's "anti-party opinion'. The documents to the Party written by the rightist Lin Xiling in 1957 imprinted a deep influence upon him... To use the terms of one of Zhang Musheng's friends: 'Now, we keep discussing about politics, but we still have not seen through as thoroughly as they had in 1957" 14 .

Their ignorance of the previous episodes of resistance, interruption of memory deprive the new generation of the rich heritage that these experiences represent. The absence of a structured memory explains that it has always been very difficult to structure an opposition movement. Because the only information about the Hundred Flowers movement came from political education classes the content of which was under absolute control of the Party, the Red Guards were unaware of the opinions that the people who were later labelled "rightists" had expressed at the time. They even denounced their "reactionary ideology". Only when they personally experienced the absolutism of Party rule did they extend the scope of their criticism to the regime as such . At this point however, they lacked intellectual resources to develop their analyses. They could not use the rightists' experiences and ideas. They had never heard of the journal Guangchang founded by Peking University (Beida) students, and of its final prohibition by the Party in May 1957 . Therefore, they had to start from scratch. They also founded "autonomous newspapers", but the language they used was far from being as sophisticated as the Rightists'.

Xu Youyu recalls how he ignored his rightist cousin's warnings. " During the Cultural Revolution, he warned me, who everyday went to post dazibao on the city walls. But I did not heed his warnings. I thought: 'your antirightist movement was nothing. This time, the supreme leader in person has liberated the 'rightist students', he has guaranteed that there would not be any 'settling of accounts after the autumn harvest'(Qiuhou 14 Li Xiaomeng, "Wenge zhong zhishiqingnian de miwang yu juexing"(Ignorance and Awareness of Rusticated Youths During the Cultural Revolution), in Ji Cilin, Niu Han, Deng Jiuping(Ed.) Women dou jingliguo de rizi, (Thedays we have been through) Beijing Shiyue wenyi chubanshe, 2000, pp. 512-3. All the titles quoted in this passage have been labeled reactionary by the authorities. The earliest, Wang Shiwei's Wild lilies and Ding Ling's "March 8 th is good" were written in 1942.

suanzhang) 15 . But, finally, I realized that myself and my generation had been duped. I went to the countryside and, when I came back to the city, three years later, I found my younger cousin busy writing dazibao, struggling his school principal, participating in the teaching revolution. I felt very sad: once a generation has seen through deception, it immediately makes way to the next generation. When I tried to understand the cause of this phenomenon, I discovered the limits of individual discourse: the lessons of our generation's experience have not entered the collective narrative, have not become part of a collective memory and high school kids that have come after us have not heard it. "16 .

Xu's reflections show the contempt in which youths of the Red Guard generation held their elders. They were all the more surprised when later, either through personal contacts or through the reading of "forbidden materials"17 , they found out that their predecessors had already covered a good part of the road. So the Peking university rightist Tan Tianrong's wish that "May 19 th [1957, the day the Democracy Wall started at Beida] and May 4 th [1919, the beginning of the New Culture movement] will remain in the minds of our younger brothers and sisters, eternally encouraging the youths who will come later"18 has not been fulfilled. In the sixties, even the most radically hostile to bureaucracy among the Red Guards 19 were in no position to use efficient political categories to further their goals.

If memory had not been controlled, one might imagine that the criticisms put up by the rightists might have helped them make their resistance more efficient. But it would take until the beginning of this century to have these discourses published, and they keep an extraordinary topicality.

In his preface to the first anthology of original writings of 1957, the famous literary critique Qian Liqun recalls that the 1957 rightists'discourse is still sort of a taboo. He notes that during the celebrations of the hundredth anniversary of the foundation of Peking University in 1998, very few people referred to that period, because "it [was] not easy to talk about it". He shares the conviction that obliteration of memory has been an important factor which has slowed the evolution of political reflection in China: "Because for a very long time (even until now), we have forgotten these precursors, these people who have sacrificed themselves! Our historians (including myself) have failed to their trade! The books written about this period of 1957 are so empty, so thin, so full of mistakes (miwu) that the 'youths who came after' (which was really the hope of the precursors) have no way to understand it. This is a shame for those who have come after, for the scholars!"20 .

THE END OF MAOÏSM AND THE RE-EMERGENCE OF MEMORY

The only way that forbidden memories can reappear and that at least part of obliterated history can make a come back into the public sphere is through a change of political line.

But this re-emergence is contained within strict guidelines. This is not, obviously, a uniquely Chinese case. In the wake of the 20 th Congress of the Communist Party of the Soviet Union, books, meetings, family reunions tackled the subject of Stalinist repression.

This was the period when Solzhenytsin published his A Day in the Life of Ivan Denissovitch, when films described the "excesses" of the Stalinist period, when writers and ordinary victims returned to their families and started to tell the stories of their lives.

But they were not allowed to question the legitimacy of communist rule.

This kind of re-emergence is not limited to communist systems either. In Chile, in Argentina (nunca mas), democratization has been accompanied by a return of the memories of the victims of repression, and the re-appraisal of past history. In Morocco since the coronation of Mohamed VI in 1999, the true history of political struggle and labor camps under Hassan II has also made its way into the public sphere, and represents an important resource for pro-democracy parties. "Behind the avalanche of historical narratives and their interpretations, one can read the fact that the State is progressively losing its monopoly on the writing of History."21

The aftermath of Mao's death was such a period, when memories which had been obliterated for a decade or more reappeared with official sanction and history was rewritten in order to grant legitimacy to Deng's line. However, there again, these were supposed to respect the master narrative elaborated by the authorities: whereas it was acceptable to denounce the "excesses" of the "gang of 4", one should never incriminate Mao Zedong or the socialist state. Besides, new narratives should follow the yiku sitian structure, and rehabilitated persons should always congratulate themselves with the excellent situation that had prevailed since the 3 rd plenum of the 11 th Party Central Committee. Many events which had not been rehabilitated could not be raised in public.

For example, as the "decision on historical mistakes" adopted by the Central Committee in 1981 stated that the anti-rightist Campaign had been too large, but was justified, it has never been possible to raise the question of its legitimacy in public, and all the books which have been published about it have followed these guidelines.

But things go even further. Periodically since the early eighties, the new leaders have declared that the whole nation should be looking towards the future in order to achieve modernization. Until now, although they themselves were victimized during the various movements, they try to prevent spontaneous acts of remembrance which take place in society. The fact that in 1987, Liu Binyan and Wang Ruowang were planning to organize a large-scale meeting in May to celebrate the people who had dared criticize the Party thirty years earlier during the Hundred Flowers campaign, was an important factor in the Centre's decision to expel them from the Party in January22 . Worse still: when in 1996, Xu

Youyu and a few friends decided to publish a book series and a journal dedicated to the memory of the Cultural Revolution on the occasion of the 30 th anniversary of the launching of the movement by Mao, they were prevented to do so by the present leadership. When a Canton periodical, Jiaodian, decided to publish a long article by Xu detailing the early events of 196623 , it was closed by the authorities. It is really hard to believe as the leadership's legitimacy since 1978 has been grounded in the denunciation of the 10 years "Cultural Revolution" but it shows that, whatever its judgement on a certain period, the Party is not ready to forsake its monopoly on the interpretation of history. It is particularly suspicious of any irruption of personal memories which might distance themselves from the official line. In fact, the controllers of ideology are convinced that if ordinary citizens are allowed to publish their views on a historical period, this will represent the first step towards the publication of documents critical of the regime.

However, in the eighties, the Party has restricted its control on the public sphere and has been obliged to forsake its grip on the private lives of its subjects. The return into society of vast numbers of people who had been jailed for political reasons, the rehabilitation of great numbers of "rightists" and "counter revolutionaries" have given rise to a huge exercise in memory inside their families and among friends. In the early eighties, whereas the leadership was exhorting the population to look toward the future (xiang qian kan), in private rooms, in crowded apartments, youths and older people were immersed in the act of remembering. Fathers told their true stories to their children, elder brothers to their younger brothers. Literary journals provided a locus for these forbidden memories to make their way into the public consciousness through poems, short stories, novels (the socalled "scars literature"). On the official scene on the other hand, Party leaders started to write their memoirs, sometimes coming back on what a journalist has called "black holes of history". Some targets of maoist political campaigns, such as Hu Feng, also wrote their memoirs in which they presented a defense of their ideas24 . More than any other field, memory is an important stake in the continuing dispute between the Party and the people.

BLACK HOLES IN CONTEMPORARY HISTORY

Despite the intensity of control, some unofficial (not recognized by the authorities)

anniversaries have become opportunities for dissenting (resistance?) forces to rally. The anniversary of the April 5th, 1976 Tian'anmen demonstration is the most obvious instance:

Whereas the leadership was trying to get rid of Zhou Enlai's political heritage, various groups opposed to Mao's policy decided to celebrate his memory on Qing Ming, the traditional Festival of the Dead. The demonstration was met with serious repression, but to many citizens this date became a landmark of resistance. In 1977 and 1978, people wishing to denounce the "leftist" line put up posters critical of both past maoist policies and the leadership of Hua Guofeng on that date. As it enjoyed a strong backing inside the apparatus, this movement was eventually victorious: on November 15 th , 1978, pressed by the need to eliminate his "whateverist" rivals, Deng Xiaoping rehabilitated the April 5 th demonstration. Later, the poems through which anonymous writers had denounced the "feudalism of the new Qin Shihuang" (a covert way to design Mao Zedong) were published officially. However, April 5th has not found a place in the official calendar of celebrations next to October 1 st , May 4 th, July 1 st and August 1 st . The fact that the demonstration was a case of rather spontaneous action by the masses probably explains the caution with which the authorities have treated it.

For one episode which has found its way into official memory, how many more are still absent from history books? First of all, the Democracy Wall movement, whereby in 1978-79, a good number of ex-Red guards expressed their criticism of the regime that had sent them to the countryside for a decade. During a few months, underground journals carrying theoretical documents which discussed the nature of the regime and the way to resolve its problems were circulated in many cities 25 . Many concerned youths, especially those who had just come back from the countryside, took part in discussions on ways to avoid the reenactment of the past tragedies. Most of them agreed on the necessity to set up autonomous organs of public opinion designed to control the Party. The articles carried in these ephemeral publications were never published on the mainland, and even now, not a single book refers to their contents. The expression Minzhu qiang (Democracy Wall) is conspicuously absent from official discourse.

But, of course, times have changed since the anti-rightist movement. Since the eighties, the Party has lost the capacity to control private discourse, and at home, people who took part in the movement can discuss their experiences with others. However, circulation of the ideas expressed during these two years has been effectively interrupted, and they are not part of the common cultural heritage of the would-be opponents 26 . In this sense, their fate was similar to that of the articles published in Guangchang in the late fifties.

As a matter of fact, students who took part in discussions about political reform in the democratic salons of the late eighties --the youths who were to take the lead of the 1989 pro-democracy movement--were not aware of the debates which had taken place a decade earlier between the radicals led by Wei Jingsheng who published Tansuo (Exploration), and moderates around Xu Wenli's Siwu luntan (April 5 th Forum) 27 . They had much more knowledge about political debates in Western countries than about the episodes of resistance which had taken place since the foundation of the People's Republic.

The absence of a structured memory prevented them to draw the lessons of the past in the field of organization: "Democracy wall" activists had tried to set up local and national associations, had founded unofficial journals and had accumulated an important experience in dealing with the authorities. They eventually fell victim to repression, but their experience was worthy. However, they could not transmit it to the younger generation who had to start back from scratches, and to create its own concepts in order to understand the workings of the system. Of course, some actors of the "Democracy Wall" were also active in the 1989 pro-democracy movement 28 , but dialogue was difficult with the students and the latter could not fully use their experience in their organizational work. Vol1, 1981, 789p. Vol.2, 1984, 611p. 27 When Wuerkaixi and other student leaders came to France, they discovered their predecessors'ideas in Widor's books.

A CHINESE PAMIATNIK?

The black out on official history and on memory of the 1989 movement also appears as a hindrance on the development of opposition. Not a single book or newspaper article published on the mainland since 1989 has ever referred to the movement in a positive light. The authorities have been very cautious and every year they are particularly tense during the two months they have termed the "sensitive period" (minggan shiqi), between 15 April and 4 June 29 . During these two months, plainclothes and uniformed policemen are particularly careful not to let anyone achieve the slightest act looking like protest, and on 4

June, --the anniversary of the massacre --the cemetery at Babaoshan is quasi under military occupation, as the Party is anxious to prevent any kind of gathering by parents who mourn the victims of the 1989 massacre.

Although history, as an officially sanctioned narrative encapsulated in books and the press, never mentions the events of 1989, the term liu si, an abbreviation reminding the officially sanctioned wu si (May Fourth) and si wu (April 4 th , 1976 the "Tian'anmen

Incident"), is extensively used by ordinary citizens as well as historians and cadres in private conversation, and, of course, overseas. This reference shows that things have changed since Mao's days, when even in personal memories political events were only referred to by their official names. When, instead of calling an event by its officially sanctioned name, ordinary citizens use the qualification given by opponents, it can be understood as a breach in the monopoly of interpretation of history by the State. During the years that immediately followed the Tian'anmen massacre, whereas the official media never raised the matter, most Peking citizens kept talking about it. In private debates among intellectuals, the analysis of the 1989 events has been a very important topic all along the nineties. Discussions about how to accelerate democratization always refer to a movement whose importance is unanimously acknowledged. However, references to "liu.si" can only be made in private or abroad. In the public sphere, the monopoly exerted by the Party remains untouched. Under these circumstances, it is hard to appreciate how much the 1989 experience has become a part of the common heritage of the Chinese citizenry.

The difficult transmission of memory has prevented the accumulation of experience by the pro-democracy movement. Every episode of resistance appears isolated to actors who view themselves as radical innovators. The 1989 students were convinced that they had been the first in the history of the PRC to have launched a real challenge to the Party, and did not view themselves as the successors of a long line of resisters (The same could be said about the Red Guard generation). This inability to capitalize on the memory of previous episodes of resistance, which is obviously also due to the Party's ban on remembrance, has resulted in a segmentation which has had a very negative impact on the structuring of a political opposition.

The history of resistance since 1949 has yet to be written (in China or abroad). So long as there is no structured memory of these past experiences, it is difficult to imagine that a well organized opposition movement will be able to emerge. Drawing lessons of the past to avoid repeating the errors is an important duty for any political organization. For if the legitimacy of a political regime is always grounded in history, opposition movements also need historical references. Especially in a country like China, where government historians (and not a few Foreign specialists) are keen to say that the search for democracy is alien to the ancient culture of this land, a detailed history of the movement of resistance to despotism would certainly provide necessary resources to the forces which fight for democracy.

The example of the Pamiatnik (memorial) movement in the late years of the Soviet Union 30 illustrates this point. In order to deprive his opponents of their historical legitimacy, and to show that communism had been much more multi-faceted than what the interpretation by Brejnev implied, Gorabachev let memory re-appear, as part of his policy of glasnost (transparency). The re-emergence of the victims' version of history, the rehabilitation of episodes of resistance were instrumental in the de-legitimization of his opponents.

China has yet to organize a "Pamiatnik" movement. Even the great writer Ba Jin's proposal to build a museum of the Cultural Revolution 31 has been rejected by the authorities. No reference to the 1989 pro-democracy movement can be found in official publications, and many victims of the various campaigns launched by Mao Zedong and his successors still expect their rehabilitation.

The request for dialogue that was at the center of the episode which took place twelve years ago has not been discussed publicly. The cyberspace has provided a locus for 29 15 th April 1989 is the day of Hu Yaobang's death which triggered the students'demonstrations. On June 4 th , the troops marched on Tian'anmen killing hundreds of unarmed protesters. 30 This organisation which asked that a memorial to the victims of stalinism be erected was founded in 1987. The word pamiat, in Russian, means "memory". dialogue, and it can be hoped that it will help dissenters and resisters to draw the lessons of past failures. In the private sphere, memory is not obliterated any more. Will it be able to batter the monopoly on the writing of history still held by the Party? Is the emergence of a Pamiatnik type movement a pre-requisite to the structuring of a Chinese opposition movement?

31 Ba Jin, "Wenge bowuguan" (A musemum of the Cultural Revolution), Suixiang lu, (Ba Jin's Random Thoughts), Beijing, Sanlian shudian, 1987, Vol.2, pp.819-823.

  25 On this episode, cf. V.Sidane, Le Printemps de Pékin (Peking Spring),Paris, Gallimard, 1981.

26

A collection of the documents from the Democracy Wall has been published in Hong Kong by Claude Widor, Documents on the Chinese Democratic

Movement :1978-1980

, Paris, Hong Kong, Editions de l'Ecole des Hautes Etudes en Sciences Sociales, the Observer Publishers,

This was the case in the communes I visited in Liaoning in 1975

This was the case in the many communes I visited as a student in Liaoning in 1976.

Xu Youyu, " Wode zaofan shenghui : Shangshan xiaxiang nao geming" (my experience of « rebellion » : up to the mountains and down to the village to make revolution) in Xu Youyu, , Ziyou de yanshuo (Free speech),Changchun, Chanchun Publishing House, 1999, p.91

ibid.p.91.

Interview with a rusticated youth fromPeking, 1989.

Xu Youyu, op.cit.., p.92.

Zhang Kangkang, "Jiti jiyi"(Collective Memory)Shouhuo,n°6, 1999.

Watson, Rubie (Ed.) Memory, History, and Opposition, Santa Fe, School of American Research Press, 1994, p.19.

In 1966 work teams sent by Liu Shaoqi to Peking universities labeled most radical Red Guards rightists or counter-revolutionaries. Later, Mao disbanded the work teams and rehabilitated the students.

Xu Youyu, « Jiyi yu shenghuo » (Memory and Life) an unpublished document sent by the author.

During the perquisitions of the early phase of the Cultural Revolution, many Red Guards stole books from the personal libraries of « stinking ninth »(Chou laojiu) e.g. intellectuals. Later, these books made their way to the countryside. Other youngsters took advantage in the chaos which reigned in the same period, to go and read " forbidden books" in public libraries.

Tan Tianrong: "Jiujiu xinling" (Save the souls) in Niu Han, Deng Jiuping (Ed.), Yuan shang cao : Jiyi zhong de fanyou yundong ((Grass on the Steppe, the Anti Rightist Campaign in Our Memory), Beijing, Jingji ribao chubanshe,

1998, p.13. 19 Cf. Yang Xiguang, "Zhongguo xiang he chu qu ?" (Whither China ?) in Révo Cul dans la Chine Pop (Cultural Revolution in Pop China),Paris, 10/18, 1972.

Qian Liqun, "An Ideological Heritage not Easy to Negate" (Bu rong mosha de sixiang yichan), in Niu Han, Deng Jiuping (Ed.), Yuan shang cao : Jiyi zhong de fanyou yundong ((Grass on the Steppe, the Anti Rightist Campaign Memory),Beijing, Jingji ribao chubanshe, 1998, p.13.

See Benjamin Stora, "Maroc, le traitement des histoires proches" (Morocco, the treatment of recent history),Esprit , 8-9, 2000, p.91.

Interview with Liu Binyan, Paris, May 1988

Youyu, "Wenhua da geming 30 nian ji" (Condolences for the 30 years of the Cultural Revolution", in Jiaodian(Focus), n° 1996, pp.10-38.

Hu Feng, "Jianshu shouhuo" (simple talk about my harvest) Excerpts in Ji Cilin, Niu Han, Deng Jiuping, Zhiiman congcongde huiyi (Memoirs in clusters of branches ) Beijing, Beijing shiyue wenyi chubanshe, pp.1-78.

See the case of Wang Juntao, the editor of the moderate underground journal Beijing zhi chun in 1978-79, who took an active part in the movement in May 1989.