Estimating the probability law of the codelength as a function of the approximation error in image compression

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1 Estimating the probability law of the odelength as a funtion of the approximation error in image ompression François Malgouyres Marh 7, 2007 Abstrat After some reolletions on ompression of images using a projetion onto a polyhedral set whih generalizes the ompression by oordinate quantization, we express, in this framework, the probability that an image is oded with K oeffiients as an expliit funtion of the approximation error. 1 Introdution In the past twenty years, many image proessing tasks have been approahed using two distint mathematial tools: image deomposition in a basis and optimization. The first mathematial approah has proved very useful and is supported by solid theoretial foundations: these guarantee its effiieny, as long as the basis aptures the information ontained in images. Modelling the image ontent by appropriate funtion spaes of infinite dimension, the mathematial theory tells us how the oordinates of an image, in a given basis, behave. For example, it is possible to haraterize Besov spaes see [12] and the spae of bounded variation whih is almost haraterized in [3] with wavelet oeffiients. As a onsequene of these haraterizations, one an obtain performane estimates for pratial algorithms see Th 9.6, pp. 386, in [11] and [5, 4] for some analyses in more omplex situations. Image ompression and restoration are the typial appliations where suh analyses are meaningful. The optimization methods whih have been applied to solve those pratial problems have also proved very effiient see [14], for a very famous example. LAGA/L2TI, Université Paris 13, 99 avenue Jean-Batiste Clément, Villetaneuse, Frane. malgouy@math.univ-paris13.fr malgouy/ 1

2 However, the theory is not able to assess how well they perform, given an image model. Interestingly, many in the ommunity who were primarily involved in the image deomposition approah are now fousing on optimization models see, for instane, the work on Basis Pursuit [2] or ompressed sensing [6]. The main reason for this is probably that optimization provides a more general framework [1, 7, 8]. The framework whih seems to allow both a good flexibility for pratial appliations see [2] and other papers on Basis Pursuit and good properties for theoretial analysis is the method of projetion onto polyhedra or polytopes. For theoretial studies, it shares simple geometrial properties with the usual image deomposition models see [10]; this should allow the derivation of approximation results. The aim of this paper is to state a theorem 1 whih relates, asymptotially as the preision grows, the approximation error and the number of oeffiients whih are oded whih we abusively all odelength, for simpliity. More preisely, when the initial datum is assumed random in a onvex set, we give the probability for the datum to be oded by K oeffiients, as a funtion of the approximation error see Theorem 3.1 for details. This result is given in a framework whih generalizes the usual oding of the quantized oeffiients non-linear approximation, as usually performed by ompression standards for instane, JPEG and JPEG Reolletion on variational ompression Here and throughout the paper N is a positive integer, I = {1,..., N} and B = ψ i i I is a basis of R N. We will also denote, for τ > 0 throughout the paper τ stands for a positive real number and for all k Z, τ k = τk 1 2. For any k i i I Z N, we set { } C k i i I = u i ψ i, i I, τ ki u i τ ki+1. 1 i I We then onsider the optimization problem { P minimize fv k i i I : under the onstraint v C k i i I, where f is a norm whih is ontinuously differentiable away from 0 and has stritly onvex level sets. In order to state Theorem 3.1, we also need f to be urved. This means that the inverse of the homeomorphism h below 2 is 1 The theorem onerning ompression in [10] is inorret. The situation turns out to be more omplex than we thought at the time that [10] was written. 2 We prove in [10] that, under the above hypotheses, h atually is an homeomorphism. 2

3 Lipshitz. h : {u R N, fu = 1} {g R N, g 2 = 1} u fu fu 2. The notation. 2 refers to the eulidean norm in R N. We denote, for any k i i I Z N, J k i i I = {i I, u i = τ ki or u i = τ ki+1}, where u = i I u i ψ i is the solution to P k i i I. The interest in these optimization problems omes from the fat that, as explained in [8], we an reover k i i I from the knowledge of J, u i j J where J = J k i i I. The problem P an therefore be used for ompression. Given a datum u = i I u iψ i R N, we onsider the unique k i u i I Z N suh that for instane i I, τ kiu u i < τ kiu+1. 2 The information J, u i j J, where J = J k i u i I, is then used to enode u. In the following, we denote the set of indexes that need to be oded to desribe u by Ju = J k i u i I. Notie that we an also show see [8] that the oding performed by the standard image proessing ompression algorithms JPEG and JPEG2000 orresponds to the above model when, for instane, f i I u i ψ i = i I u i Observe that the above ompression sheme works for any quantization table see [8]; we restrit to the uniform quantization beause Theorem 3.1 only applies in this ontext. However, several levels of generalization are possible, if one wants to generalize it to more general quantization tables. Notie that, in the theorem, we assume that the data belong to a given level set, denoted L fd τ, of a norm f d. Therefore, the ode attributed to eah oeffiient need not to be infinite. 3 The estimate Theorem 3.1 Let τ > 0 and U be a random variable whose low is uniform in L fd τ, for a norm f d. Assume f satisfies the hypotheses given in Setion 2. For any norm. and any K {1,... N} there exists D K suh that for all ε > 0, there exists T > 0 suh that for all τ < T P # J U = K D K E N K N+1 + ε, 3

4 where E is the approximation error 3 : E = E U τ k i Uψ i. i I Moreover, if f i I u iψ i = i I u i 2 1 2, we also have 4 P # J U = K D K E N K N+1 ε. The proof of the above theorem is given in [9]. Its two main steps are: the haraterization of all the k i i I L fd τ whih are oded with K oeffiients, for any given K {1,... N}; the ensus, for eah K, of k i i I obtained at the first step. When the above theorem differs from the results evoked in Setion 1 in several ways. First, it onerns variational models whih are more general than the model for whih the results of Setion 1 are usually stated. This is probably the main interest of the urrent result. For instane, by a reasoning similar to the one used in the proof of Theorem 3.1, it is probably possible to obtain approximation results for redundant transforms. Seondly, it expresses the distribution of the number of oeffiients as a funtion of the approximation error, whereas earlier results do the opposite. Typially, they bound the approximation error quantified by the L 2 norm by a funtion of the number of oeffiients that are oded. The advantages and drawbaks of the different kinds of statements is not very lear. In the framework of Theorem 3.1, the larger D K for K small, the better the model ompresses the data. However, it is lear that, as the approximation error goes to 0, it is more and more likely to obtain a ode of size N. In this respet, the onstant D N 1 seems to play an important role, sine it dominates asymptotially as τ goes to 0 the probability not to obtain a ode of length N. Thirdly, the theorem is stated for data leaving in a finite dimension vetor spae and, as a onsequene, it does not impose a priori links between the data distribution the funtion f d and the model the funtion f and the basis B. The ability of the model to represent the data is always assessed by the C K. Of ourse, an analogue of Theorem 3.1 for data leaving in infinite dimension spae would be interesting. Referenes [1] A. Chambolle, R.A. De Vore, N. Lee, and B.J. Luier. Nonlinear wavelet image proessing: Variational problems, ompression and noise removal 3 When omputing the approximation error, we onsider the enter of C k i i I has been hosen to represent all the elements u suh that k i u i I = k i i I. 4 This assumption is very pessimisti. For instane, the lower bound seems to hold for almost every basis B of R N, when f is fixed. However, we have not worked out the details of the proof of suh a statement. 4

5 through wavelet shrinkage. IEEE, Transations on Image Proessing, 73: , Speial Issue on Partial Differential Equations and Geometry-Driven Diffusion in Image Proessing and Analysis. [2] S. S. Chen, D. L. Donoho, and M. A. Saunders. Atomi deomposition by basis pursuit. SIAM Journal on Sientifi Computing, 201:33 61, [3] A. Cohen, W. Dahmen, I. Daubehies, and R. De Vore. Harmoni analysis of the spae bv. Revista Matematia Iberoameriana, 19: , [4] A. Cohen, I. Daubehies, O.G. Guleryus, and M.T. Orhard. On the importane of ombining wavelet-based nonlinear approximation with oding strategies. IEEE, Transations on Information Theory, 487: , July [5] A. Cohen, R. De Vore, P. Petrushev, and H. Xu. Nonlinear approximation and the spae bv. Amerian Journal of Mathematis, 1213: , June [6] D. Donoho. Compressed sensing. IEEE, Trans. on Information Theory, 524: , April [7] F. Malgouyres. Minimizing the total variation under a general onvex onstraint for image restoration. IEEE, Trans. on Image Proessing, 1112: , De [8] F. Malgouyres. Image ompression through a projetion onto a polyhedral set. Tehnial Report , University Paris 13, August [9] F. Malgouyres. Estimating the probability law of the odelength as a funtion of the approximation error in image ompression. Tehnial Report sd , CCSD, Otober [10] F. Malgouyres. Projeting onto a polytope simplifies data distributions. Tehnial Report , University Paris 13, January [11] S. Mallat. A Wavelet Tour of Signal Proessing. Aademi Press, Boston, [12] Y. Meyer. Ondelettes et opérateurs, volume 1. Hermann Ed., [13] R.T. Rokafellar. Convex analysis. Prineton University Press, [14] L. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. Physia D, 60: ,

6 4 Proof of Theorem First properties and reolletion Rewriting P For any u R N, P u denotes the optimization problem { minimize fv u P u : under the onstraint v C 0, where 0 denotes the origin in Z N and C. is defined by 1. We then denote, for any u = i I u iψ i C 0, Ju = {i I, u i = τ 2 or u i = τ 2 }. With this notation, the set of ative onstraints of the solution u to P u is simply Ju. Proposition 4.1 For any k i i I Z N J k i i I = Ju, where u is the solution to P τ i I k iψ i. Proof. Denoting ũ the solution of P k i i I and u the solution to P τ i I k iψ i, we have ũ = u + τ k i ψ i. 3 i I This an be seen from the fat that P i I k iψ i is exatly P ki i I, modulo a global translation by τ i I k iψ i. The rigorous proof of 3 an easily be established using Kuhn-Tuker onditions, see [13], Th 28.3, pp The proposition is then obtained by identifying the oordinates of ũ and u in the basis B On projetion onto polytopes We an now adapt the definitions and notations of [10] to the problems P.. Beside Proposition 4.6, all the results stated in this setion are proved in [10]. We onsider a norm f d whih will be used latter on to define the data distribution law and define for any C R N and any A R S A C = { u R N, u C, u is solution to P u and f d u u A }. This orresponds to all the optimization problems whose solution is in C we also ontrol the distane between u and the result of P u. Notie that S A C 6

7 depends on τ. We do not make this dependene expliit sine it does not reate any onfusion, in pratie. We also define the equivalene relationship over C0 u v Ju = Jv. For any u C0, we denote u the equivalene lass of u. In the ontext of this paper, we obviously have for all u = i I u iψ i C0 u = u + τ β j ψ j, j Ju, 1 2 < β j < 1 2, 4 where j Ju u = j Ju u j ψ j. Here and all along the paper the notation j J stands for j I \ J. Let us give some desriptions of S... Proposition 4.2 For any u C0 and any v u, In words, S 1 v is a translation of S 1 u. S 1 v = v u + S 1 u. Proposition 4.3 For any u C0, any v S ]0,+ [ u and any λ > 0 u + λv u S ]0,+ [ u. Theorem 4.4 For any u C0 and any τ > 0, S ]0,τ ] u = { v + λu u, for v u, λ ]0, τ ] and u S 1 u } We also have see [10] Proposition 4.5 If f satisfies the hypotheses given in Setion 2, for any u C0, S 1 u is a non-empty, ompat Lipshitz manifold of dimension #Ju 1. Another useful result for the purpose of this paper is the following. Proposition 4.6 If f satisfies the hypotheses given in Setion 2, for any u C0 and any τ > 0, S ]0,τ ] u is a non-empty, bounded Lipshitz manifold of dimension #Ju. Proof. In order to prove the proposition, we onsider u = i I u i ψ i C0 and u = i Ju u i ψ i. We are going to prove the proposition in the partiular ase where u = u. Proposition 4.2 and 4.3 permit indeed to generalize the latter result obtained to any S ]0,τ ] u S ]0,τ ] u is obtained by translating S]0,τ ] u., for u u. They indeed guarantee that 7

8 In order to prove that S ]0,τ ] u is a bounded Lipshitz manifold of dimension #Ju, we prove that the mapping h defined below is a Lipshitz homeomorphism. h : S 1 u ]0, τ ] S ]0,τ ] u u, λ u + λu u. The onlusion then diretly follows from Proposition 4.5. Notie first that we an dedue from Proposition 4.3, that h is properly defined. Let us prove that h is invertible. For this purpose, we onsider λ 1 and λ 2 in ]0, τ ] and u 1 and u 2 in Su 1 suh that We have 5 u + λ 1 u 1 u = u + λ 2 u 2 u. 6 λ 1 = f d λ 1 u 1 u = f d λ 2 u 2 u = λ 2. Using 6, we also obtain u 1 = u 2 and h is invertible. Finally, h is Lipshitz sine, for any λ 1 and λ 2 in ]0, τ ] and any u 1 and u 2 in S 1 u, λ 1 u 1 u λ 2 u 2 u 2 = λ 1 u 1 u 2 + λ 1 λ 2 u 2 u 2, where C is suh that for all u S 1 u, τ u 1 u C λ 1 λ 2, u u 2 C. Remember Su 1 is ompat, see Proposition The estimate We denote the disrete grid by D = {τ k i ψ i, k i i I Z N }, i I and, for u C0 and k j j Ju Z Ju, D k j j Ju = {τ j Ju k j ψ j +τ The set D k j j Ju is a slie in D. i Ju k i ψ i, where k i i Ju Z I\Ju }. 8

9 Proposition 4.7 Let τ > 0, u C0 and k j j Ju Z Ju, # S ]0,τ ] D k u j j Ju 1. Proof. Taking the notations of the proposition and assuming S ]0,τ ] D k u j j Ju, we onsider ki 1 i I and ki 2 i I suh that τ i I ki 1 ψ i S ]0,τ ] D k u j j Ju and τ i I ki 2 ψ i S ]0,τ ] D k u j j Ju. Theorem 4.4 guarantees there exist v 1 and v 2 in u, λ 1 and λ 2 in ]0, τ ] and u 1 and u 2 in Su 1 suh that τ ki 1 ψ i = v 1 + λ 1 u 1 u i I and τ i I k 2 i ψ i = v 2 + λ 2 u 2 u. So v 1 + λ 1 u 1 u = v 2 + λ 2 u 2 u + τ ki 1 k2 i ψ i. i Ju Using 4, we know there exists β 1 i i Ju and β 2 i i Ju suh that and i Ju, 1 2 < β1 i < 1 2 and 1 2 < β2 i < 1 2, v 1 = u + τ v 2 = u + τ i Ju i Ju β 1 i ψ i β 2 i ψ i, with u = j Ju u j ψ j, where u = i I u i ψ i. So, letting for all i Ju, α i = ki 1 k2 i + β2 i β1 i, we finally have λ 1 u 1 u = λ 2 u 2 u + τ α i ψ i. 7 i Ju Let us assume max α i > 0, 8 i Ju 9

10 and onsider 0 < λ 1 suh that We have, using 7, λ < 1 2 max i Ju α i. 9 u + λλ 1 [u 1 u + u u ] = u + λλ 1 u 1 u = u + λτ α i ψ i + λλ 2 u 2 u i Ju = v + λλ 2 [u 2 u + v v], where v = u + λτ i Ju α iψ i. Moreover, using 4 and 9, we know that v u. Using Proposition 4.2, we know that u 1 u + u S 1 u and u 2 u + v S 1 v. Finally, applying Theorem 4.4, we obtain u + λλ 1 u 1 u S ]0,τ ] u S]0,τ ] v. Sine the solution to P u +λλ 1 u 1 u is unique, we neessarily have u = v and therefore max i Ju α i = 0. This ontradits 8 and guarantees that max α i = 0. i Ju Using the definition of α i, we obtain, for all i Ju, k 1 i k2 i = β1 i β2 i < 1. This implies ki 1 = k2 i, for all i I. Let us denote, for u C0, the projetion onto Span ψ j, j Ju by p : R N Span ψ j, j Ju i I α iψ i j Ju α jψ j. It is not diffiult to see that, for any τ > 0, u C0 and k j j Ju Z Ju, # S ]0,τ ] D k u j j Ju = 1 = τ k j ψ j p S ]0,τ ]. 10 u j Ju Remark 1 Notie that the onverse impliation does not hold in general. It is indeed possible to build ounter examples where S ]0,τ ] passes between the points of u the disrete grid D. However, it is not diffiult to see that, if τ j Ju k jψ j p S ]0,τ ] and S ]0,τ ] D k u u j j Ju =, we an build ki i Ju Z J\Ju suh that τ k j ψ j + τ k i ψ i S ]0,τ ] j Ju i Ju u, 10

11 where u = u jψ j.u j are the oordinates of u j Ju This means that the set S ]0,τ ] u, whih is a manifold of dimension #Ju living in R N, intersets a disrete grid. This is obviously a very rare event. Typially, adding to the basis B some kind of randomness for instane adding a very small Gaussian noise to every ψ i would make it an event of probability 0. Notie, with this regard, that when f i I u iψ i = i I u i 2, we trivially have the equivalene in 10. A simple onsequene of 10 is that # S ]0,τ ] D # p S ]0,τ ] u u τ j Ju k j ψ j, k j j Ju Z Ju. 11 Notie finally that, for u = i I u i ψ i C0, Proposition 4.2 and Equation 4 guarantees that p S 1 u = p S 1 u, for u = j Ju u j ψ j. We therefore have, using also Theorem 4.4, Proposition 4.3 and Equation 4, p S ]0,τ ] = {pv + λpu pu, for v u, λ ]0, τ ] and u S 1 u u }, Finally, # S ]0,τ ] D u = {u + λpu u, for λ ]0, τ ] and u Su 1 }, = p S ]0,τ ] u. # p u τ S ]0,τ ] j Ju k j ψ j, k j j Ju Z Ju. 12 u Proposition 4.8 If f satisfies the hypotheses given in Setion 2 then, for any u = i I u i ψ i C0, p S ]0,τ ] where u = j Ju u j ψ j is a nonempty, bounded Lipshitz manifold of dimension #Ju. Proof. Thanks to Proposition 4.6, it suffies to establish that the restrition of p : p : S ]0,τ ] u p S ]0,τ ] u u pu. is a Lipshitz homeomorphism. This latter result is immediate one we have established that p is invertible. 11

12 This proof is similar to the one of Proposition 4.7. Taking the notations of the proposition, we assume that there exist u 1 and u 2 in S ]0,τ ] u and α i i Ju R Ju satisfying u 1 = u 2 + τ α i ψ i. i Ju If we assume max i Ju α i 0, we have for 0 < λ < min1, 1 2 max i Ju α i, u + λu 1 u = u + τ i Ju = v + λ u 2 + τ λα i ψ i + λu 2 u i Ju λα i ψ i v for v = u +τ i Ju λα iψ i. Sine v u see 4, Proposition 4.2 guarantees that u 2 + τ i Ju λα iψ i = u 2 + v u S v ]0,τ]. As a onsequene, applying Proposition 4.3, we know that u + λu 1 u S λ u S]0,+ [ v. Sine P u + λu 1 u has a unique solution, we obtain a ontradition and an onlude that for all i Ju, max i Ju α i = 0. As a onsequene, p is invertible. It is then obviously a Lipshitz homeomorphism. Proposition 4.8 guarantees that p S ]0,τ ] u is Lebesgue measurable in R #Ju. p S ]0,τ ] u is Moreover, its Lebesgue measure in R #Ju denoted L #Ju finite and stritly positive : 0 < L #Ju p S ]0,τ ] u <. Another onsequene takes the form of the following proposition. Proposition 4.9 Let τ > 0 and u C0 lim τ K # S ]0,τ ] D L τ 0 u K p S ]0,τ ] u where K = #Ju. Moreover, if the equality holds in 11 or equivalently : the equality holds in 12 lim τ K # S ]0,τ ] D = L τ 0 u K p S ]0,τ ] u. 12

13 Proof. In order to prove the proposition, we are going to prove that, denoting K = #Ju, lim τ K # p S ]0,τ ] u τ 0 τ k j ψ j, k j j Ju Z Ju = L K p S ]0,τ ] u j Ju The onlusion follows from 12. Let us first remark that, unlike S ]0,τ ] u, the set A = p S ]0,τ ] u u 13 does not depend on τ. This is due to Proposition 9 5, in [10]. Notie also that, beause of Proposition 4.8, both A and p S ]0,τ ] u are Lebesgue measurable in R K and that L K A = L K p S ]0,τ ] u. In order to prove the upper bound in 13, we onsider the sequene of funtions, defined over R K f n u = max 0, 1 n inf u v 2. v A This is a sequene of funtions whih are both Lebesgue and Riemann integrable and the sequene onverges in L 1 R K to 1 A the indiator funtion of the set A. So, for any ε > 0, there exists n N suh that f n 1 A + ε. Moreover, we have, for all u R K and all n N, 1 A u f n u. { So, denoting V τ = τ } j Ju k jψ j u, k j j Ju Z Ju, lim # p S ]0,τ ] u τ 0 τ k j ψ j, k j j Ju Z Ju j Ju = lim τ K 1 A v τ 0 v V τ lim τ K f n v τ 0 v V τ f n 1 A + ε L K p S ]0,τ ] u + ε. 5 The definition of SC A given in the urrent paper does not allow the rewriting of the proposition 9 of [10]. This is why we have not adapted it in Setion

14 So, lim τ K # p S ]0,τ ] u τ 0 τ j Ju k j ψ j, k j j Ju Z Ju L K p S ]0,τ ] u The lower bound in 13 is obtained in a similar way, by onsidering an approximation of 1 A by a funtion smaller than 1 A whih is Riemann integrable. For instane : f n u = 1 max 0, 1 n inf v A u v 2. From now on, we will denote for all K {1,..., N} C K = τ u j ψ j, where J I, #J = K and j J, u j = 1 2 or u j = 1 2 j J The set C K ontains all the enters of the equivalene lasses of odimension K. Similarly, we denote = {u C0, #Ju = K}. We obviously have, for all K {1,..., N}, = u C K u. Sine, for all K {1,..., N}, C K is finite, it is lear from Proposition 4.9 that, for any τ > 0, lim τ K # S ]0,τ ] τ 0 D L K p S ]0,τ ] u < + u C K Moreover, we have an equality between the above two terms, as soon as the equality holds in 11. We an finally express the following estimate. Proposition 4.10 Let τ > 0 lim τ K # S ]0, [ τ 0 L fd τ D u C K L K p S ]0,τ ] u where K = #Ju. Moreover, if the equality holds in 11 for all u C K or equivalently : the equality holds in 12 lim τ K # S ]0, [ τ 0 L fd τ D = u C K L K p S ]0,τ ] u. 14

15 Proof. We onsider M = We have, for all u C0, So sup f {u= P d u i I uiψi, i I, ui 1 2 } f d u Mτ. 14 We therefore have for all u L fd τ and for u the solution to P u, f d u u f d u + f d u S ]0, [ τ + Mτ. L fd τ S ]0,τ +Mτ]. Moreover, it is not diffiult to see that remember h defined by 5 is an homeomorphism lim L K p S ]0,τ +Mτ] u = L τ 0 K p S ]0,τ ] u. u C K u C K We an therefore dedue from Proposition 4.9 that lim τ K # S ]0, [ τ 0 L fd τ D u C K L K p S ]0,τ ] u In order to prove the last statement of the proposition, we onsider u C0 and u S ]0,τ ], we know that u f d u f d u u + f d u τ + Mτ So Sine again lim L K τ 0 u C K S ]0,τ Mτ] S ]0, [ L fd τ. p S ]0,τ Mτ] u = L K p S ]0,τ ] u, u C K we know that the seond statement of the proposition holds. Another immediate result is useful to state the final theorem. Notie first that we have, for any k i i I Z N and any norm., v τ k i ψ i dv = Cτ N+1, v Ck i i I i I 15

16 where C = {v= P v dv i I viψi, i I, vi 1 2 } only depends on the partiular norm. and the basis ψ i i I. So, denoting U a random variable whose law is uniform in L fd τ and k i U i I the disrete point defined by 2, we have E U τ i I lim k iuψ i τ 0 τ N+1 = C. 15 This follows from the fat that the number of points k i i I suh that Ck i i I intersets both L fd τ and its omplement in R N beomes negligible with regard to the number of points k i i I suh that Ck i i I is inluded in L fd τ, when τ goes to 0. We an now state the final result. Theorem 4.11 Let τ > 0 and U be a random variable whose low is uniform in L fd τ, for a norm f d. For any norm., any K {1,... N} and any ε > 0, there exists T > 0 suh that for all τ < T P # J U = K where E is the approximation error 6 : E = E U τ i I D K E N K N+1 + ε, k i Uψ i, Moreover, if the equality holds in 11 or equivalently : the equality holds in 12 for all u C K, then we also have P # J U = K D K E N K N+1 ε. The onstant D K is given by with and B = A K = D K = u C K L K A K BC N K N+1, p S ]0,τ ] u, L N L fd τ L N {v = i I v iψ i, i I, v i 1 2 } C = {v= P v dv. i I viψi, i I, vi 1 2 } 6 When omputing the approximation error, we onsider the enter of C k i i I has been hosen to represent all the elements oded by P k i i I. 16

17 Proof. Remark first that, for any k i i I Z N, the probability that τ ki U i τ ki+1, when U = i I U iψ i follows a uniform law in L fd τ, is L N Ck i i I L fd τ L N L fd τ. Therefore, taking the notation of the theorem P # JU = K = k i i I Z N 1 τ P i I kiψi S[0,+ [ L N Ck i i I L fd τ L N L fd τ. If k i i I is suh that L N Ck i i I L fd τ 0, there exists v C0 suh that v + τ i I k iψ i L fd τ. So, we have f d τ i I k i ψ i τ + f d v τ + Mτ, where M is given by 14. We therefore have P # JU = K L N C0 L N L fd τ # S ]0,+ [ L fd τ + Mτ D. The lower bound is obtained with a similar estimation and we obtain P # JU = K L N C0 L N L fd τ # S ]0,+ [ L fd τ Mτ D. Notie finally that lim τ 0 # # S ]0,+ [ L fd τ D S ]0,+ [ L fd τ ± Mτ D = 1. The proof is now a straightforward onsequene of Proposition 4.10 and 15. More preisely, taking the notations of the theorem and ε > 0, we know that there exists T > 0 suh that, for all τ < T, τ K # S ]0, [ L fd τ + Mτ D A K + ε, and E 1 N+1 C 1 N+1 τ ε. 17

18 So P # J K i i I = K τ N B A K + ε τ K A K + ε B A K BC N K N+1 E C 1 N+1 + ε N K E N K N+1 + o1, where o1 is a funtion of ε whih goes to 0, when ε goes to 0. The first inequality of the theorem follows. The proof of the seond inequality of the theorem is similar to one above. 18