A Ventura

N Antheaume

ENVIRONMENTAL ASSESSMENT OF CARBON CAPTURE AND UTILIZATION: A NEW SYSTEMIC VISION -APPLICATION TO VALORIZATION OF NICKEL SLAGS

With increasing social awareness, concerning impacts of climate change, carbon capture and utilization (CCU) is gaining traction on a world scale. It is being pushed forward by the European Union as one of the important levers to mitigate its greenhouse gases emissions. Life Cycle Assessment (LCA), and its derived carbon footprint method, is thus becoming a crucial point of support to provide a strategic political vision. However, these guidelines rely on current LCA standards and practices and are not adapted to provide a strategic vision. System modelling of environmental impacts of valorization technologies with LCA use substitution scenarios and/or consequential LCA. Consequential LCA only considers unconstrained markets: every product issued from a valorization or CCU process is always considered as a substitute for a determining product. However, this is not always what is observed, especially in the construction and demolition sector. Valorization technologies are always developed according to a given state of external constraints defined by waste composition and amounts as well as existing regulations and their amounts and nature is constantly changing. Thus, the first focus of waste treatment industries is thus to find possible technological means to "get rid of" undesirable amounts of waste or carbon dioxide, but the fact that it can replace existing and determining products is by and large ignored. To reflect actual situation, there is a need to introduce the possibility of constrained markets in the models. With constrained markets, price elasticity cannot be used as the basic relationship between price, demand and produced amounts. Thus, for constrained markets, the product scale is not adapted. To overcome the abovementioned obstacles, we propose a transdisciplinary approach described in this paper. This approach is currently being developed through its application to valorization of New Caledonia nickel slags by CCU, in the CARBOSCORIES 2 and CARBOVAL projects.

INTRODUCTION

With increasing social awareness, concerning impacts of climate change, carbon capture and utilization (CCU) is gaining traction on a world scale. It is being pushed forward by the European Union as one of the important levers to mitigate its greenhouse gases emissions.

Life Cycle Assessment (LCA), and its derived carbon footprint method, is thus becoming a crucial point of support to provide a strategic political vision. A guideline document has been recently published [START_REF]Techno-Economic Assessment & Life-Cycle Assessment Guidelines for CO2 Utilization[END_REF] to provide decision making support for CCU, promoting a combination of two methods: Technological and Economic Assessment (TEA) and LCA, followed by a Multi-Criteria Decision Analysis.

However, these guidelines rely on current LCA standards and practices and are not adapted to provide a strategic vision as it will be further explained in the following part of this paper.

The purpose of this article is to detail scientific obstacles to current LCA practices, to propose a new methodological approach and to outline its application to the CARBOVAL project.

ANALYSIS OF SCIENTIFIC OBSTACLES

Modelling requirements for LCA assessment of CCU technologies are very similar to modelling waste valorization using LCA models. For CCU, specific technologies are being developed to store or valorize carbon dioxide from air and industrial flue gases. This is similar to waste treatment, for which dedicated technologies are developed to store or valorize waste produced by various human activities. In both cases, these technologies aim at storing or valorizing a given material object (either carbon dioxide or waste), because it is deemed undesirable. The word "undesirable" means negative effects on society and/or the environment. For instance, carbon dioxide contributes to climate change, incineration of waste cause air pollution, and landfill of waste use potentially valuable land and are a possible source of air, water and soil pollution.

The European Union considers that developing waste valorization technologies is important to reduce: global warming for CCU; land occupation and pollution for waste because turning them into useful products also becomes economically worth treating them. Eventually, existing products, often produced using natural resources and fossil resources, can be substituted by valorized waste. The combined environmental and economic benefits that result constitute the virtuous cycle of a circular economy.

Accordingly, both CCU and waste valorization rely upon technological processes that combine at least two objectives: "treating" (to be understood mostly as "getting rid of") an undesirable material object, and "producing a usable and desirable product" from it. In LCA, these valorization processes can thus have two possible functions that will lead to two different choices for the reference flow [START_REF] Ekvall | The Application of Life Cycle Assessment to Integrated Solid Waste Management: Part 2-Perspectives on Energy and Material Recovery from Paper[END_REF] as shown in Figure 1. The "getting rid of" objective reflects a function of undesirable matter management that will lead to the amount of undesirable object as an input reference flow. The "producing a usable and desirable good" objective relates to a given functionality provided by the desirable product, which will lead to the amount of product as an output reference flow. Several models correspond to this type of evaluation, but most current ones are based on substitution scenarios and consequential LCA. An abundant literature is available on both methods. They both originate in the late 1990's [START_REF] Weidema | Marginal production technologies for life cycle inventories[END_REF] and were later improved by their initial author [START_REF] Weidema | Avoiding Co-Product Allocation in Life-Cycle Assessment[END_REF]- [START_REF] Weidema | Avoiding Co-Product Allocation in Life-Cycle Assessment[END_REF] as well as by many case studies leading to various methodological improvements. In consequential LCA, a determining product is a product whose production is driven by market demand, whereas a co-product is a product whose production depends on the production of the determining product. The market of a co-product is called a "constraint" market, as its production is not driven by demand but by the production of a determining product. Consequential LCA method use market displacement models; these are based on the price elasticity generated by the market displacement that results from valorized product introduction into the existing market. However, price elasticity models can only be applied to unconstrained markets (because they rely upon law of supply and demand), and consequential LCA thus only considers unconstrained markets [START_REF] Brandão | Consequential Life Cycle Assessment: What, How, and Why ?[END_REF]. In other words, every product issued from a valorization or CCU process is always considered as a substitute for a determining product. However, this is not always what is observed. In the construction and demolition sector, cement concrete demolition waste can potentially be used as recycled concrete aggregates (RCA) and thus as a substitute for aggregates produced by a quarry [START_REF] Mousavi | Territorial Environmental Modeling of Cement Concrete Demolition Waste (CCDW) Management with a Life Cycle Approach[END_REF]. More generally, valorization

Producing system

Using system processing technologies are always developed according to a given state of external constraints defined by waste composition and amounts as well as existing regulations [START_REF] Grimaud | Conception des scénarios de recyclage pilotée par l'évaluation des performances de procédés[END_REF].

Because the amounts and nature of undesirable material objects is constantly changing [START_REF] Grimaud | Conception des scénarios de recyclage pilotée par l'évaluation des performances de procédés[END_REF], the first focus of waste treatment industries is thus to find possible technological means to "get rid of" undesirable amounts of material objects, but the fact that it can replace existing and determining products is by and large ignored.

To reflect actual situations, there is a need to introduce the possibility of constrained markets in the models. With constrained markets, price elasticity cannot be used as the basic relationship between price, demand and produced amounts. Thus, for constrained markets, the product scale is not adapted.

An illustration is given in Figure 2: ten units of a valorized product are produced and replace 100% of an existing product. In that example, the existing product contributes 200 units of an impact (i.e. Global Warming Potential), whereas the valorized product (from CCU or waste valorization) only contributes 100.

At the product scale, let us assume for the example that a classical consequential LCA model based on price elasticity with unconstrained markets can result in a 100% substitution. One will thus find that one unit of a valorized product can reduce the impacts by 50% compared to one unit of a substituted product. This is the possible result that can be obtained by either substitution scenarios or consequential LCA with a 100% substitution.

However, for constrained markets, the balance between the mass flows of valorized and substituted products must be considered. Figure 2 depicts three possibilities. In market A, market volume is important, i.e. 1,000 units of products in that example. With the 100% substitution scenario, the valorized product gains a market share that corresponds the 10 produced units; the impacts are then reduced by 5%. In market B, market volume is small, but the valorized product is still competitive. As the amount of valorized product is constrained, 10 units are produced, but the constrained market of substituted products only requires 1 such unit. The current product is then substituted by the valorized product, avoiding 200 impact units, but the 9 other units of valorized product are still produced because constrained amounts are determined by the need "to get rid of" and not by the market demand. The overall impact will then increase by 300%, and 9 units of valorized product will have to be stocked. Finally, in the case of market C, market volume is small and the valorized product is not competitive. If valorized product is still being produced (again as per the need "to get rid of" and not by the market demand), the overall impact jumps 500% as current product is not avoided, and 10 units of valorized product must then be stocked.

Flow matching between valorized and substituted products, coupled with market analysis and modeling are all necessary to derive an acceptable strategic political vision.

PROPOSED METHODOLOGY

To overcome the abovementioned obstacles, we propose the approach illustrated in Figure 3.

Step 1 relates to the process model of the CCU technology, including modelling of inventory flows for LCA and cost calculations (OPEX and CAPEX).

Step 2 investigates possible usage scenarios of the valorized product. These scenarios embed the product characteristics (physical, chemical, sensorial…), which can depend on both the waste properties (in the case of waste valorization processes only) and the operating parameters of the CCU technology. Expected functionalities of the valorized product ought to match current or expected market requirements, as defined by market research. Indeed, whereas engineering studies target functionalities, only usage can validate them [START_REF] Ventura | Conclusion of the workshop[END_REF].

Step 3 is an analysis of competitiveness of the valorized product in comparison with current products for the targeted market. In step 4, a market model is used to predict the total amount substituted by the change induced from introducing the valorized product in the market. This model is developed at the market scale (as previously explained from Figure 2), and thus integrates actual market volumes in a given geographical area according to Material Flow Analysis (MFA) studies. In step 5, both the nature and amount of substituted products are identified, so that LCA of the multi-product system can be conducted. As all steps from 1 to 6 are parametrized and interconnected, a detailed sensitivity analysis can be conducted to identify improvement levers for all actors of the system and both environmental impacts and competitiveness, following previously developed methodologies [START_REF] Ventura | Action-oriented Life Cycle Assessment: example on hemp-based construction materials[END_REF]- [START_REF] Senga Kiessé | Introducing economic actors and their possibilities for action in LCA using sensitivity analysis: Application to hemp-based insulation products for building applications[END_REF]. The proposed method could thus correspond to a system expansion, for which expansion is based on equivalent functionalities of products. Functional units of products are a key issue in step 2, 3 and 4. In step 2, they can be defined using current LCA methodology, but analysis of market requirements is necessary to validate the choice of substituted products. With the global multi-products system, the expanded system is used to establish a strategic political vision; it is not oriented on a given product, and provides improvements at country and/or global scales.

The burden sharing between products is not a crucial point for a strategical political vision, which requires assessing environmental improvements at country and/or global scales. With this objective in mind, the question of allocation methods is not a key issue. However, this can be a key issue for companies that are responsible for products involved in the overall studied system. Furthermore, many of the technologies involved can be based on industrial symbiosis, especially when environmental performance is a target. However, the choice of burden sharing rules cannot be based on scientific evidence, as it is also a political choice [START_REF] Chen | LCA allocation procedure used as an incitative method for waste recycling: An application to mineral additions in concrete[END_REF]. It could be tested as an additional parameter in the modelled system in order to favour the most efficient cooperation between actors towards environmental improvements. Moreover, efficient cooperation for industrial symbiosis is very dependent on local conditions and we are of the opinion that it should be up to the cooperating partners to define and agree upon the sharing burdens rules through private contracts: they will necessarily reach an agreement that is financially positive for all of them.

FUTURE APPLICATION: VALORIZATION OF NICKEL SLAGS BY CCU

A research consortium consisting of LGC, CNRT and BRGM, three French research institutes based in metropolitan France and New Caledonia have previously established an innovative CCU mineralization solution [START_REF] Julcour | Development of an attrition-leaching hybrid process for direct aqueous mineral carbonation[END_REF]. This technology is currently being further developed with the objective of valorizing pyrometallurgical Nickel slags produced in New Caledonia into valuable construction products (hydraulic binders, pozzolanes, …). Using such products in the construction sector could be highly beneficial from several standpoints, including preservation of natural resources, greenhouse gas mitigation, and reduction of land use for slag disposal. The latter is particular critical in the constrained environment of the New-Caledonian territory, a 19,000 km2 island in the southwest Pacific Ocean.

It goes without saying that the actual environmental benefits of the Ni slag valorization project, through the proposed mineralization process and possible product valorization routes, must be assessed. This endeavor is one of cornerstones of the CARBOSCORIES 2 and CARBOVAL projects, which are funded by the French Environment and Energy Management Agency (ADEME) and the Caledonian Energy Agency (ACE). These projects bring together a multidisciplinary research team that includes several institutes, namely LGC, CNRT, BRGM, LMDC, IFSTTAR, ESO and LEMNA. The transdisciplinary approach described in Figure 3 is coordinated by IRSTV, a research federation dedicated to urban science and technology [START_REF] Baud | HOME IRSTV[END_REF]. The actual contributors from IRSTV are IFSTTAR for the LCA component of the approach, LEMNA for the economic analysis and associated models, and ESO for the urban metabolism vision. The IRSTV team will develop and implement the approach depicted in Figure 3 to the Nickel slags valorization project using mineralization. The context of the project will provide the IRSTV team with the perfect case study for implementing, improving and testing the approach discussed in this paper. In addition to guiding the research team towards viable valorization routes for Nickel slags, the research work will develop a methodology that is expected to be transferable to other situations where CCU, through mineralization for example, is considered as a pathway for waste valorization.

CONCLUSION

The basis for a transdisciplinary approach have been set to respond to issues of environmental assessment of CCU and waste valorization. Although based on consequential Life Cycle Assessment, this new approach differs from it as the model: is based on multi-products system, considers market scale, and includes possibility of constrained market models. Still, it has to be tested on actual case studies.

Figure 8 .

 8 Figure 8. The two possible reference flows of a waste valorization or CCU process in LCA In addition, the possible environmental benefits of these technologies can only be evaluated by comparison with both what is avoided when not valorizing carbon dioxide or waste, and with what is substituted by the valorized product.

Figure 9 .

 9 Figure 9. Effects on environmental impacts of a change induced by a valorized product (from waste or CCU), at market scale versus product scale

etFigure 10 .

 10 Figure 10. Transdisciplinary approach for environmental assessment of CCU technologies

Proceedings of the International Workshop CO2 Storage in Concrete -CO2STO2019France -Marne La Vallée -Ifsttar -June 24-25, 2019

ACKNOWLEDGMENTS

The French Environment and Energy Management Agency (ADEME) and the New Caledonia Energy Agency (ACE) fund this research through the CARBOVAL and CARBOSCORIES 2 projects.


Zhiqing Kui

Jean Baccou
email: jean.baccou@irsn.fr

Jacques Liandrat
email: jacques.liandrat@centrale-marseille.fr

ON THE CONSTRUCTION OF MULTIRESOLUTION ANALYSES ASSOCIATED TO GENERAL SUBDIVISION SCHEMES

Keywords: Multiresolution, Subdivision schemes, Detail operators, Decimation schemes, Non-linear schemes

Subdivision schemes are widely used in numerical mathematics such as signal/image approximation, analysis and control of data or numerical analysis. However, to develop their full power, subdivision schemes should be incorporated into a multiresolution analysis that, mimicking wavelet analyses, provides a multi-scale decomposition of a function, a curve, or a surface. The ingredients needed to define a multiresolution analysis associated to a subdivision scheme are a decimation scheme and detail operators. Their construction is not straightforward as soon as the subdivision scheme is non-interpolatory.

This paper is devoted to the construction of decimation schemes and detail operators compatible with general subdivision schemes, including non-linear ones. Analysis of the performances of the constructed analyses is carried out. Some numerical applications are presented in the framework of image approximation.

Introduction

Subdivision schemes have been introduced and developed in many fields, leading to specific schemes targeting smoothness, stability, shape preserving, manifold preserving and many other properties.

The complete framework into which subdivision schemes can develop their powerful properties is the multiresolution framework that, mimicking the wavelet framework, involves two other ingredients : a decimation operator and the detail operators. When subdivision schemes (only binary subdivision scheme will be considered in this paper) are uniform, interpolatory, linear and stationary, the decimation operator h and detail operators (g, g) are easily defined. This is not the case as soon as the subdivision scheme is more general (e.g. [START_REF] Dyn | Univariate subdivision and multi-scale transforms: The nonlinear case, Multiscale, Nonlinear and Adaptive Approximation[END_REF]).

This paper is devoted to a generic construction of multiresolution frameworks for general subdivision schemes. The originality of the constructions relies on the development of a systematic approach that can be used to generate new types of multiresolution analysis associated to non-standard subdivision schemes. In terms of applications, these developments should allow the design of new frameworks for the decomposition and reconstruction of functions, signals and images.

The paper is organized as follows : we first provide an overview of the multiresolution framework associated to subdivision schemes (Section 2). Two examples of schemes for which no associated multiresolution existed are also given to emphasize the main motivation of our work. Then starting from first results presented in [START_REF] Kui | On the coupling of decimation operator with subdivision schemes for multi-scale analysis[END_REF], we focus on the construction of decimation operators consistent with general subdivision schemes (Section 3). Section 4 is devoted to the construction of detail operators while the analysis of linear and non-linear multiresolutions is provided in section [START_REF] Arandiga | Approximation of piecewise smooth functions and images by edge-adapted (eno-ea) nonlinear multiresolution techniques[END_REF]. Different examples and applications are detailed in the last two sections.

2. Overview on subdivision schemes as basic elements of the multiresolution framework 2.1. Subdivision schemes. A binary subdivision operator h [START_REF] Dyn | Subdivision schemes in computer aided geometric design[END_REF] is defined through a real-valued sequence (h k ) k∈Z having a finite number of non-zero values such that

(f k ) k∈Z ∈ l ∞ (Z) → ((hf ) k ) k∈Z ∈ l ∞ (Z) with (hf ) k = ∑ l∈Z h k-2l f l .
Any set {h k : k 0 ≤ k ≤ k 1 , k ∈ Z} containing all the non-zero values of (h k ) k∈Z is called a mask of the operator h of length k 1 -k 0 + 1 and is denoted M h . Subdivision is generally iterated starting from an initial sequence (f j0 k ) k∈Z to generate (f j k ) k∈Z , j > j 0 as

f j+1 = hf j . ( 1
)
At each scale j, the polygon (k2 -j , f j k ), k ∈ Z is considered. The advantage of using subdivision for data prediction relies on the flexibility in the choice of the mask. The simplest strategy that consists in considering the same mask for every position, scale and data f j (leading to linear uniform and stationary operators) [START_REF] Deslauriers | Symmetric iterative interpolation processes[END_REF] can be improved by introducing a dependence on k (non-uniform scheme) [START_REF] Cohen | Quasilinear subdivision schemes with applications to ENO interpolation[END_REF] or j (non-stationary scheme) [START_REF] Beccari | An interpolating 4-point C 2 ternary nonstationary subdivision scheme with tension control[END_REF]. The linearity can be also relaxed by taking into account the values of the sequence (f j l ) l∈Z for l in the vicinity of k [START_REF]On a nonlinear subdivision scheme avoiding Gibbs oscillations and converging towards C s functions with s > 1[END_REF]. The analysis of a subdivision scheme relies on the study of two important properties related to its convergence and stability. These notions are recalled in the following definition where || • || denotes any norm. When required, || • || ∞ stands for the infinity norm of sequences (l ∞ norm) or functions (L ∞ norm).

Definition 1 (Convergence of a subdivision scheme). A subdivision scheme is uniformly

convergent if ∀f ∈ l ∞ (Z), ∃f ∞ ∈ C 0 (R), such that lim j→∞ ||h j f -f ∞ (2 -j •)|| ∞ = 0.

Definition 2 (Stability of a subdivision scheme). A convergent subdivision scheme

h is stable in the norm || • || if there exists a constant C ∈ R such that for all f, f ϵ ∈ l ∞ (Z), ∀i ∈ N, ||h i f -h i f ϵ || ≤ C||f -f ϵ ||.
Note that if the subdivision h is linear, the stability (∃C ∈ R, ∀i ∈ N, ||h i || ≤ C) is a direct consequence of the convergence of the subdivision scheme.

Multiresolution transforms.

In the multiresolution framework, if (V j ) j∈Z denotes a family of separable spaces with f j ∈ V j , relation (1) can be exploited to define the prediction from V j to V j+1 ([14], [START_REF] Arandiga | Multiresolution based on weighted averages of the hat function I: linear reconstruction techniques[END_REF], [START_REF]Multiresolution based on weighted averages of the hat function II: non-linear reconstruction techniques[END_REF]). The connection from fine to coarse levels is performed thanks to a decimation operator denoted h. If f j+1 ∈ V j+1 is obtained after subdivision i.e. f j+1 = hf j , a decimation must satisfy the following consistency condition in order to recover f j from f j+1 :

Definition 3 (Consistency property). A decimation operator h is said to be consistent with the subdivision scheme h if hh = I

where I stands for the identity operator.

However, if f j ∈ V j is obtained after decimation of f j+1 ∈ V j+1 , hf j usually does not coincide with f j+1 . Thus, a prediction error is introduced as :

(2)

e j+1 = f j+1 -h hf j+1 .
The relation f j = hf j+1 associated to formula (2) defines a mapping from V j+1 to V j × V j+1 sending f j+1 to {f j , e j+1 }. A left inverse is given by f j+1 = hf j + e j+1 . This mapping can be reduced to a bijective mapping from V j+1 to V j × W j with W j a suitable subspace of V j+1 . This leads to introduce g, a detail decimation operator and its right inverse g :

W j → V j+1 . The quantity d j = gf j+1 is called a detail.
The quadruplet of operators (h, h, g, g) fully defines a multiresolution associated to the subdivision scheme h and satisfies the relation

h h + gg = I.
This quadruplet is said to be compatible if and only if, moreover, hh = I, gg = I, hg = 0 and gh = 0.

Iterating, for a fixed level j 0 ≤ j, the multi-scale decomposition of a sequence f j is the element {f j0 , d j0 , d j0+1 , ..., d j-1 }. Similarly a reconstruction transform can be introduced to recover f j from {f j0 , d j0 , d j0+1 , ..., d j-1 }.

The performance of the multiresolution process is controlled by the behavior of the prediction error and more precisely by the amount of small error values at each scale. Generally, the norm of the prediction error exponentially decays with the scale j, and the decay rate plays a key role in the sparsity of the multi-scale decomposition. The definition of the decay rate is recalled as follows.

Definition 4 (Decay of the norm of the prediction error). A prediction error e j is said to decay with a decay rate p in the norm || • || if and only if for all f j ∈ l ∞ (Z), there exists a constant C independent on j such that

∀j ′ < j, ||e j ′ || ≤ C2 -pj ′ .
Since truncation is a key ingredient in image approximation algorithms, the stability has to be considered. In addition to Definition 2, this leads to the following stability definitions.

Definition 5 (Stability of a decimation scheme). A decimation scheme

h is stable in the norm || • || if there exists a constant C ∈ R such that for all f, f ϵ ∈ l ∞ (Z), ∀i ∈ N, || hi f -hi f ϵ || ≤ C||f -f ϵ ||.
||f j -f j ϵ || ≤ C(||f j0 -f j0 ϵ || + j-1 ∑ i=j0 ||d i -d i ϵ ||).
The decomposition transform is said to be stable in the norm || . || if there exists a constant C such that for all (f j , f j ϵ ),

||f j0 -f j0 ϵ || + j-1 ∑ i=j0 ||d i -d i ϵ || ≤ C||f j -f j ϵ ||.
The multiresolution is said to be stable if the associated decomposition transform and reconstruction transform are stable.

Subdivision-based multiresolution is an appealing framework for data analysis since it inherits the flexibility of the construction of subdivision operators. It has been considered as an interesting alternative to classical orthogonal or biorthogonal wavelet approaches in several scientific fields such as image processing [START_REF] Arandiga | Approximation of piecewise smooth functions and images by edge-adapted (eno-ea) nonlinear multiresolution techniques[END_REF] or computational mathematics [START_REF] Charina | Multigrid methods: Grid transfer operators and subdivision schemes[END_REF]. However, the construction of the full multiresolution process is not straightforward when the subdivision scheme is non-interpolatory 1 or non-linear. The contributions of this paper are developped to specifically tackle this problem. To illustrate the typical situations addressed by our work, we first recall two types of schemes that will be further studied in Section 6 and emphasize the open issues concerning the construction of the associated multiresolution.

2.3.

Examples of subdivision schemes. Two specific schemes are described in this section. The first one is the 4-point shifted Lagrange subdivision scheme [START_REF] Dyn | A C 2 four-point subdivision scheme with fourth order accuracy and its extensions[END_REF]. It is non-interpolatory and is defined by }.

     f j+1 2k = - 7 128 f j k-1 + 105 128 f j k + 35 128 f j k+1 - 5 128 f j k+2 , f j+1 2k+1 = - 5 128 f j k-1 + 35 128 f j k + 105 128 f j k+1 - 7 128 f j k+2 . The mask associated to this subdivision scheme is M h = {h k }, -4 ≤ k ≤ 3 with (3) M h = {-
The second one is the non-interpolatory and non-linear 4-point shifted PPH scheme [START_REF]On a nonlinear subdivision scheme avoiding Gibbs oscillations and converging towards C s functions with s > 1[END_REF]. Introducing

∆ 2 f k = (f j k+1 -f j k ) -(f j k -f j k-1 ), it is defined as : if |∆ 2 f k | ≤ |∆ 2 f k+1 |, (4) { f j+1 2k = -7 128 f j k-1 + 105 128 f j k + 35 128 f j k+1 -5 128 f j k+2 -5 64 DHA k , f j+1 2k+1 = -5 128 f j k-1 + 35 128 f j k + 105 128 f j k+1 -7 128 f j k+2 -7 64 DHA k , if |∆ 2 f k | > |∆ 2 f k+1 |, (5) { f j+1 2k = -7 128 f j k-1 + 105 128 f j k + 35 128 f j k+1 -5 128 f j k+2 -7 64 DHA k , f j+1 2k+1 = -5 128 f j k-1 + 35 128 f j k + 105 128 f j k+1 -7 128 f j k+2 -5 64 DHA k ,
, 1 A subdivision scheme h is said to be interpolatory if ∀f, ∀k, (hf ) 2k = f k where

DHA k = H(∆ 2 f k , ∆ 2 f k+1 ) -A(∆ 2 f k , ∆ 2 f k+1 ), A(x, y) = x + y 2 , H(x, y) = xy x + y
(sign(xy) + 1), and sign(x, y) =

{ -1, if xy < 0 1, if xy ≥ 0 ,
Note that the 4-point shifted PPH scheme can be considered as the linear 4-point shifted Lagrange scheme with a non-linear perturbation.

Both schemes are non-interpolatory. They are therefore well suited to take into account noise in the data. Moreover, the PPH approach is data-dependent. It leads to accurate prediction of data exhibiting non-regular behavior. However, up to now,the efficiency of these schemes within the multiresolution framework has been ignored since no multiresolution framework (i.e. the operators h, g and g) was available.

The two following sections are therefore devoted to the introduction of generic methods to derive h and (g, g) for such general subdivision operators.

Before ending this section, we introduce some notations that will be used in the sequel :

• A linear decimation operator h is defined through a real-valued sequence ( hk ) k∈Z having a finite number of non-zero values such that (

f k ) k∈Z ∈ l ∞ (Z) → (( hf ) k ) k∈Z ∈ l ∞ (Z) with ( hf ) k = ∑ l∈Z hl-2k f l .
Any set of the form { hk : k 0 ≤ k ≤ k 1 , k ∈ Z} containing all non-zero values of ( hk ) k∈Z is called a mask of the operator h of length k 1 -k 0 + 1 and is denoted M h.

• For any operator h (resp. h) defined through the real value sequence (h k ) k∈Z (resp. ( hk ) k∈Z ) and any integer t ∈ N, the translated operator T t (h) (resp. T t ( h)) is defined through the sequence (h k-t ) k∈Z (resp.

( hk-t ) k∈Z ). • We call (σ, σ ′ ) the pair of subsampling operators defined as

∀e ∈ l ∞ (Z), { (σe) k = e 2k+1 , (σ ′ e) k = e 2k , k ∈ Z.
and τ the interlacing operator,

∀u, v ∈ l ∞ (Z), (τ (u, v)) k = { u i , k = 2i + 1, v i , k = 2i, i ∈ Z.
• We denote he and ho the two operators associated to even and odd terms of a linear decimation operator hL ,

∀k ∈ Z, f ∈ l ∞ (Z),        ( ho f ) k = ∑ l hL 2l+1-2k f 2l+1 ( he f ) k = ∑ l hL 2l-2k f 2l .

Construction of consistent decimation operators

In the sequel, the subdivision h is fixed but general.

3.1. Linear case. A generic method to construct the mask of any decimation consistent with a given linear uniform subdivision has been proposed in [START_REF] Kui | On the coupling of decimation operator with subdivision schemes for multi-scale analysis[END_REF]. The main results are recalled in the two following propositions. The first one is devoted to the construction of a finite family of elementary operators while the second one describes how all consistent decimation operators can be recovered using linear combinations of translated versions of elementary operators.

Proposition 1.

Let h be a linear subdivision operator whose mask is constructed from the sequence

{h n-2α , h n-2α+1 , . . . , h n , h n+1 }.
Introducing,

H M h =            h n h n-2 • • • h n-2α 0 • • • 0 h n+1 h n-1 • • • h n-2α+1 0 • • • 0 0 h n h n-2 • • • h n-2α • • • 0 0 h n+1 h n-1 • • • h n-2α+1 • • • 0 . . . . . . 0 0 • • • h n h n-2 • • • h n-2α 0 0 • • • h n+1 h n-1 • • • h n-2α+1            , if det(H M h ) ̸ = 0
, there exists 2α consistent elementary decimation operators whose masks are of length 2α. These masks are given by each row of H -1 M h .

Proposition 2. The subdivision operator h being fixed and satisfying the hypotheses of Proposition 1, let { h(i) } 1≤i≤2α be the set of elementary consistent decimation operators. Then, any consistent decimation operator can be constructed as

∑ t∈T ∑ i∈I c i,t T 2t ( h(i) ) (6) with ∀t ∈ T ⊂ Z, ∑ i∈I c i,t = δ t,0 , and 0 ∈ T .
The large choice of decimation masks generated by this approach is of prime importance in practice since it allows to tune some specific characteristics of the scheme according to given objectives.

The key ingredient in this construction of linear consistent decimation operators is the translation invariance of the subdivision mask. In the case of non-linear schemes, since this property is lost, the method introduced above is not tractable. This situation is studied in the next section.

Non-linear case.

Introducing (h L , hL ) a couple of consistent linear subdivision and decimation operators, the following proposition provides a generic method to derive a non-linear decimation operator h consistent with a non-linear subdivision scheme h. Proposition 3. Let h be a subdivision operator. If there exists a linear decimation hL so that hL h -I is contractive in the l ∞ norm 2 , then for any f j+1 ∈ l ∞ (Z), the fixed-point equation

(7) f j = hL f j+1 -( hL h -I)f j
has a unique solution. Moreover, h : f j+1 → f j is a decimation operator consistent with h.

Proof. The existence and uniqueness of the solution of the fixed-point equation is a consequence of the Banach fixed point theorem since hL h -I is contractive.

Starting from f j+1 such that f j+1 = h f j , it is clear that f j is a solution of equation [START_REF] Charina | Multigrid methods: Grid transfer operators and subdivision schemes[END_REF]. Since the solution is unique, f j is the decimated sequence from f j+1 and ( 7) defines a consistent decimation. □ Remark 1.

(1) Given f j+1 , the different choices of linear decimation operators hL lead to different f j and associated prediction errors. This flexibility is exploited in the numerical tests of Section 7.

(2) If hL is consistent with h, then hL h = I and the fixed-point equation reduces to f j = hL f j+1 .

Construction of detail operators

In this section the construction of the multiresolution transforms is completed exhibiting a couple of operators (g, g) compatible with (h, h).

For linear subdivision schemes, the prediction error (2) belongs, by construction, to the kernel of the associated linear consistent decimation operator. This statement guarantees the existence of a couple of detail subdivision and detail decimation operators (g, g) which are compatible with the couple of subdivision and decimation operators (h, h).

A similar result can be derived for general subdivision and decimation constructed following Proposition 3. It is given by the next proposition, Proposition 4. Let h be a general subdivision operator and h be a consistent decimation operator given by Proposition 3 with hL the involved linear decimation operator. The associated prediction error e j+1 satisfies [START_REF] Cohen | Biorthogonal bases of compactly supported wavelets[END_REF] hL e j+1 = 0, and (9) he j+1 = 0.

Proof. Under the contraction condition, the unique solution for equation ( 7) is denoted f j = hf j+1 . Then, the prediction error can be written as

e j+1 = f j+1 -h hf j+1 = f j+1 -h L f j -(h -h L ) f j = (I -h L hL )f j+1 -(I -h L hL )(h -h L ) f j .
2 The operator U is said to be contractive in the norm

|| • || if there exists c ∈ R, 0 < c < 1, such that for all (u, v), ||U u -U v|| ≤ c||u -v||.
Applying hL and using the consistency relation leads to [START_REF] Cohen | Biorthogonal bases of compactly supported wavelets[END_REF]. Let w j = he j+1 . Then

w j = hL e j+1 -hL (h -h L )w j = -( hL h -I)w j .
According to the fixed-point theorem, w j = 0 which leads to [START_REF] Cohen | Quasilinear subdivision schemes with applications to ENO interpolation[END_REF]. □

Then the detail operators can be constructed as follows, Proposition 5. Let h be a subdivision operator and h a consistent decimation constructed following Proposition 3. If there exists a linear left inverse operator of he , denoted ( he ) -1 , then (g, g) defined as

(10) { g = σ(I -h h) g = τ (•, -( he ) -1 ho •)
are detail operators compatible with (h, h).

Proof. Thanks to Proposition 4, we have hL e j+1 = 0 that can be written as ho σe j+1 + he σ ′ e j+1 = 0. Taking d j = σe j+1 ,we get σ ′ e j+1 = -( he ) -1ho d j . Therefore, e j+1 = τ (σe j+1 , σ ′ e j+1 ) = τ (d j , -( he ) -1ho d j ). Detail operators then read { d j = σe j+1 = gf j+1 e j+1 = gd j where (g, g) are defined by [START_REF] Deslauriers | Symmetric iterative interpolation processes[END_REF]. Finally, we prove the compatibility of (h, h, g, g) : According to Proposition 4, we have hL g = 0 and hg = 0. Then gg = σ(Ih h)g = σg -σh hg = σg = I.

Under the consistency condition hh = I, we have gh = σ(I -h h)h = 0, which concludes the compatibility proof. □ Remark 2. If h is an interpolatory subdivision, the subsampling operator h defined by ∀f, ∀k, ( hf ) k = f 2k is consistent with h and ho stands for the identity operator.

Analysis

Here, we investigate the stability and the prediction error decay of linear and non-linear multiresolutions. In the first case, the stability is ensured as soon as the subdivision and the decimation are stable. It is not true in the non-linear framework where extra conditions are required (see [START_REF] Amat | Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms[END_REF] for example in the case of non-linear interpolatory schemes). However, we restrict in the next section the stability analysis to the subdivision and decimation operators. We refer to Section 7.1 for a numerical evidence of the stability of the non-linear shifted PPH multiresolution transforms.

Stability of the subdivision and of the decimation.

5.1.1. Linear case. In the linear case, the stability of the subdivision is a consequence of its convergence. We focus on the stability study of the decimation operator.

The following proposition provides a necessary and sufficient condition for stability. Proposition 6. A linear decimation operator h is stable if and only if the subdivision of mask constructed from the sequence 2( hl ) l∈Z is stable.

Proof. If f j = h j f 0 , then

f j l = ∑ lj-1 h l-2lj-1 ∑ lj-2 h lj-1-2lj-2 • • • ∑ l1 h l2-2l1 ∑ l0 h l1-2l0 f 0 l0 = ∑ l0 h j l-2 j l0 f 0 l0 .
The operator h is stable if and only if there exists C ∈ R, such that ∀l,

∑ l0 h j l-2 j l0 ≤ C. Since ∑ l h j l-2 j l0 is independent of l 0 , then (11) ∀l 0 , ∑ l h j l-2 j l0 ≤ 2 j C.
If f 0 = hj f j , then

f 0 l0 = ∑ l1 hl1-2l0 ∑ l2 hl2-2l1 • • • ∑ lj-1 hlj-1-2lj-2 ∑ lj hlj-2lj-1 f j l = ∑ l hj l-2 j l0 f j l . ( 12
)
If the subdivision is constructed from the sequence 2( hl ) l∈Z , ( 11) is equivalent to

∑ l hj l-2 j l0 = 2 -j ∑ l h j l-2 j l0 ≤ C,
which is exactly the stability condition for the decimation h. □

The assumption on the stability of the subdivision associated to 2( hl ) l∈Z can be replaced by a general condition that can be easier to verify in practice. Let us first recall without proof the following useful lemma, Lemma 1. The decimation operator h is stable if and only if, for some i ∈ N * , hi is stable. Proposition 7. The decimation operator h is stable if and only if there exists i ∈ N * , such that the subdivision h constructed from the sequence 2( hi l ) l∈Z is stable.

Proof. The proof is straightforward by taking j = ik in the proof of Proposition 6, since [START_REF] Dyn | A C 2 four-point subdivision scheme with fourth order accuracy and its extensions[END_REF] can be rewritten as

f 0 l0 = ∑ l1 hl1-2l0 ∑ l2 hl2-2l1 • • • ∑ l ik-1 hl ik-1 -2l ik-2 ∑ l ik hl ik -2l ik-1 f ik l ik = ∑ li ( ∑ li-1 • • • ∑ l1 hli-2li-1 • • • hl1-2l0 ) • • • ∑ l ik ( ∑ l ik-1 • • • ∑ l ik-i+1 hl ik -2l ik-1 • • • hl ik-i+1 -2l ik-i )f ik l ik = ∑ li hi li-2 i l0 • • • ∑ l ik hi l ik -2 i l ik-i f ik l ik .
Lemma 1 provides the conclusion. □ 5.1.2. Non-linear case. In the non-linear case, there already exist some results for the stability of the subdivision [START_REF] Amat | Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms[END_REF]. We focus again on the decimation.

Proposition 8.

As above, we suppose that h = h L + h N and note hL a stable decimation consistent with h L such that hL h N is µ 1 < 1 Lipschitz and

( I + hL h N ) -1 is β 0 Lipschitz. Let µ k , k > 1, denote the Lipschitz constant of ( hL ) k h N .
If there exists L > 1 and α < 1 such that for all k ≥ L, µ k ≤ α k then, the non-linear decimation defined trough the fixed point equation ( 7) is stable as soon as the constant C µ2,...,µ L ,α,β0 < 1

where

C µ2,...,µ L ,α,β0 = β 0 ( ∑ L-1 k=1 µ k+1 + α L+1 1-α
) .

Proof. Iterating equation [START_REF] Charina | Multigrid methods: Grid transfer operators and subdivision schemes[END_REF] one gets that for all p > 0, f j-p = ( hL ) p+1 f j+1 -∑ p k=0 ( hL ) k+1 h N f j-p+k . It follows that for any couple of sequences (f j+1 , g j+1 ), if we note δ l = ||f l -g l ||, we get, using the stability of hL and the Lipschitz hypotheses,

δ j ≤ β 0 Cδ j+1 , ∀p ≥ 1, δ j-p ≤ β 0 Cδ j+1 + β 0 ∑ p k=1 µ k+1 δ j-p+k , ≤ β 0 Cδ j+1 + β 0 ( ∑ L-1 k=1 µ k+1 δ j-p+k + ∑ p k=L α k+1 δ j-p+k
) .

If we call s p = max{δ j-k , -1 ≤ k ≤ p}, we get that for all p ≥ 0,

s p ≤ β 0 Cs -1 + β 0 ( L-1 ∑ k=1 µ k+1 + α L+1 -α p+2 1 -α ) s p-1 .
Since α < 1, as soon as C µ2,...,µ L ,α,β0 < 1, then, s p is bounded by C ′ s -1 . That concludes the proof. □

Decay of the prediction error.

In the linear case, the decay of the prediction error for consistent scheme (h, h) was studied in [START_REF] Kui | On the coupling of decimation operator with subdivision schemes for multi-scale analysis[END_REF] and is recalled in the following proposition, Proposition 9. Let h be a linear subdivision scheme such that ∀n = 0, 1, 2, . . . , p -1,

∑ l∈Z h l l n (-1) l = 0,
then for any consistent linear decimation h, the decay rate (defined in Definition 4) in the l ∞ norm of the associated prediction error is at least p.

Remark 3. Since the derivation of Proposition 9 relies on the Taylor expansion, this theoretical decay rate is significant only when the data are regular.

This result can be generalized to the non-linear case as follows :

Proposition 10. Let h be a non-linear subdivision scheme with h = h L +h N where h L verifies Proposition 9 and such that for all f j ∈ l ∞ (Z), there exists a constant C independent on j such that

(13) ||h N f j || ∞ ≤ C2 -q(j+1) .
If h is a stable ( in the l ∞ norm) and consistent decimation operator constructed according to Proposition 3, then the decay rate of the associated prediction error is at least min(p, q).

Proof. For a consistent couple (h, h) deduced by Proposition 3,

f j = hf j+1 = hL f j+1 -hL h N f j .
The associated prediction error can be written as

e j+1 = (I -h L hL )f j+1 -(I -h L hL )h N f j .
Since (I -h LhL ) is a linear operator and h is stable, there exists constants C 1 and C 2 such that

||e j+1 || ≤ C 1 2 -p(j+1) + C 2 2 -q(j+1) ≤ (C 1 + C 2 )2 -min(p,q)(j+1) ,
and the decay rate is at least min(p, q). □

Examples

We focus in this section on the linear and non-linear multiresolutions associated to the subdivision schemes recalled in Section 2.3. For each of them, the construction of a consistent decimation is addressed. Exploiting the results established in the previous section, a stability analysis is performed and the prediction error decay of the multiresolutions is studied as well.

4-point Shifted Lagrange Scheme. Construction of consistent linear decimations.

Applying Proposition 1, we get 6 elementary decimations whose masks are given by the rows of HM h with

HM h =       
        .
With equation ( 6), a symmetrical consistent decimation operator h of length 8 can be formed by combining the 3rd and 4th rows of HM h as ( 14) This operator is used in the next section to construct a consistent decimation associated to the non-linear scheme PPH.

M h = { h-4 , h-3 , h-2 , h-1 , h0 , h1 , h2 , h3 } = {
Stability. The shifted Lagrange scheme has been proven to be convergent (and therefore stable) in [START_REF] Dyn | A C 2 four-point subdivision scheme with fourth order accuracy and its extensions[END_REF]. We have moreover Proposition 11. The decimation operator given by ( 14) is stable.

Proof. Considering the subdivision operator constructed from 2( h3 l ) l∈Z where h is given by ( 14), a direct calculation gives ∀i 1 , i 2 , . . . , i 9 ∈ {0, 1}, max

( ( 1 2 ) 9 ||A (1) i1 A (1) i2 • • • A (1) i9 || ) = 0.98334 < 1,
where A

(1) 0 , A

(1) 1

are associated refinement matrices for differences. According to [START_REF] Dyn | Subdivision schemes in computer aided geometric design[END_REF], this proves that the subdivision 2( h3 l ) l∈Z is convergent. Therefore, h is stable thanks to Proposition 7. □ Remark 4. The subdivision related to the sequence 2( hl ) l∈Z with h given by ( 14) is not convergent since there exists eigenvalue of the associated refinement matrices larger than 1. The proposed pair of consistent and stable subdivision/decimation operators is therefore out of the range covered by biorthogonal multiresolutions described in [START_REF] Cohen | Biorthogonal bases of compactly supported wavelets[END_REF].

Decay of the prediction error.

A direct calculation provides that the mask (3) satisfies Proposition 9 with p = 5. Therefore, the decay rate of the prediction error associated to the 4-point shifted Lagrange subdivision and any consistent decimation scheme is 5.

4-point Shifted PPH Scheme.

To derive a consistent decimation operator for this scheme, following Proposition 4, a linear decimation operator hL consistent with the linear part of the subdivision scheme h L and such that hL h N is contractive should be constructed. Before introducing the main result, two lemmas are required:

Lemma 2. ∀x, y, a, b ∈ R, |(H(x, y) -A(x, y)) -(H(a, b) -A(a, b))| ≤ 2 • max(|x -a|, |y -b|).
Proof. we distinguish different cases:

(1) x, a > 0, y, b < 0, then H(x, y) = H(a, b) = 0, |(H(x, y) -A(x, y)) -(H(a, b) -A(a, b))| ≤|A(x, y) -A(a, b)| ≤max(|x -a|, |y -b|). (2) x, y, a > 0, b < 0, then H(a, b) = 0, |(H(x, y) -A(x, y)) -(H(a, b) -A(a, b))| = a -x 2 + b -y 2 + 2xy x + y ≤ a -x 2 + b 2 + y 2 x -y x + y + 1 2 2xy x + y ≤ 1 2 |x -a| + 1 2 |y -b| + max(|x|, |y|) ≤2 • max(|x -a|, |y -b|). (3) x, y, a, b > 0 or x, y > 0, a, b < 0, |(H(x, y) -A(x, y)) -(H(a, b) -A(a, b))| = ( 2xa (x + y)(a + b) - 1 2 )(y -b) + ( 2yb (x + y)(a + b) - 1 2 )(x -a) ≤max ( 2xa + 2yb (x + y)(a + b) -1 , 2xa -2yb (x + y)(a + b) ) • max(|x -a|, |y -b|) ≤2 • max(|x -a|, |y -b|).
and we can conclude the other cases by symmetry.

□ Lemma 3. Given x, y, a, b ∈ R, if (|x| -|y|)(|a| -|b|) < 0, then ∀p, q ∈ R, |p(H(x, y) -A(x, y)) -q(H(a, b) -A(a, b))| ≤ max(|p|, |q|) • max(|x -a|, |y -b|).
Proof. We first consider p = q = 1,

|(H(x, y) -

|(H(x, y) -A(x, y)) -(H(a, b) -A(a, b))| ≤ x -a 2 + y -b 2 ≤max(|x -a|, |y -b|). (2) xy > 0, ab < 0 then H(a, b) = 0, |(H(x, y) -A(x, y)) -(H(a, b) -A(a, b))| ≤ x -y 2 + a + b 2 ≤max(|x -a|, |y -b|). (3) xy > 0, ab > 0, |(H(x, y) -A(x, y)) -(H(a, b) -A(a, b))| = xa + xb + ya -yb 2(x + y)(a + b) (x -a) + -xa + xb + ya + yb 2(x + y)(a + b) (y -b) ≤ 1 2 |x -a| + 1 2 |y -b| ≤max(|x -a|, |y -b|).
and we can conclude the other cases by symmetry.

In the other hand, (|x| -|y|)(|a| -|b|) < 0 leads to

|(H(x, y) -A(x, y)) + (H(a, b) -A(a, b))| ≤ max(|x -a|, |y -b|),
and

|H(x, y) -A(x, y)| ≤ max(|x -a|, |y -b|).
Then for p, q ∈ R satisfying, (1) p > q > 0,

|p(H(x, y) -A(x, y)) -q(H(a, b) -A(a, b))| =|q(H(x, y) -A(x, y)) -q(H(a, b) -A(a, b)) + (p -q)(H(x, y) -A(x, y))| ≤(|q| + |p -q|) • max(|x -a|, |y -b|) ≤p • max(|x -a|, |y -b|).
(

) p > 0 > q, |p| > |q|, |p(H(x, y) -A(x, y)) -q(H(a, b) -A(a, b))| =| -q(H(x, y) -A(x, y)) -q(H(a, b) -A(a, b)) + (p + q)(H(x, y) -A(x, y))| ≤(|q| + |p + q|) • max(|x -a|, |y -b|) ≤p • max(|x -a|, |y -b|). 2
and we can conclude the other cases by symmetry. □ Construction of consistent non-linear decimations. We have:

Proposition 12. Writing the 4-point shifted PPH subdivision scheme as h = h L + h N with h L given by ( 3) and taking hL given by ( 14), then hL h N is contractive. Therefore, equation ( 7) defines a consistent decimation.

Proof. In order to prove the contractivity, we focus on

|| hL h N u -hL h N v||. Let's denote p = 2L 2 ( 1 4 ) = 2L -1 ( 3 4 ) = -5 64 and q = 2L 2 ( 3 4 ) = 2L -1 ( 1 4 ) = -7 64 . (1) if |∆ 2 u k | ≤ |∆ 2 u k+1 | and |∆ 2 v k | ≤ |∆ 2 v k+1 |, |( hL h N u) l -( hL h N v) l | ≤ ∑ k p • h2k-2l + q • h2k+1-2l |DHA u k -DHA v k |.
(

) if |∆ 2 u k | > |∆ 2 u k+1 | and |∆ 2 v k | > |∆ 2 v k+1 |, |( hL h N u) l -( hL h N v) l | ≤ ∑ k q • h2k-2l + p • h2k+1-2l |DHA u k -DHA v k |. (3) if (|∆ 2 u k | -|∆ 2 u k+1 |)(|∆ 2 v k | -|∆ 2 v k+1 |) < 0, according to Lemma 3, |( hL h N u) l -( hL h N v) l | ≤ ∑ k max ( |p • h2k-2l + q • h2k+1-2l |, |q • h2k-2l + p • h2k+1-2l | ) • max(|∆ 2 u k+1 -∆ 2 v k+1 |, |∆ 2 u k -∆ 2 v k |). Since |DHA u k -DHA v k | ≤ 8 • ||u -v||, max(|∆ 2 u k+1 -∆ 2 v k+1 |, |∆ 2 u k -∆ 2 v k |) ≤ 4 • ||u -v||, 2
combining the previous cases with ( 14) leads to

|| hL h N u -hL h N v|| ≤ 307 384 ||u -v||.

□

Stability. The shifted PPH scheme has been proven to be stable in [START_REF]On a nonlinear subdivision scheme avoiding Gibbs oscillations and converging towards C s functions with s > 1[END_REF]. Concerning the decimation, Proposition 8 requires evaluating the Lipschitz constants of several operators involved in the non-linear decimation. These constants are numerically estimated from a set of sequences constructed by random perturbations (10 realizations of a gaussian noise, N (0, 10)) of the 260th column of the man image depicted on Figure 2. They are provided on Table 1. 1. Numerical estimation of the Lipschitz constants associated to the operators involved in the non-linear decimation.

Operators Lipschitz constant ( I + hh N ) -1 β 0 = 1.12 hL h N µ 1 = 0.34 ( hL ) 2 h N µ 2 = 0.22 ( hL ) 3 h N µ 3 = 0.16 ( hL ) 4 h N µ 4 = 0.1 ( hL ) 5 h N µ 5 = 0.07 {( hL ) k h N } k≥6 α = 0.67 Table
It follows that C µ2,...,µ5,α,β0 = 0.92 that leads to the stability of the non-linear decimation.

Decay of the prediction error. According to Proposition 10, and assuming that the decimation is stable, a direct calculation provides that ( 13) is satisfied with q = 4 for PPH scheme. Therefore, the decay rate of the prediction error associated to the 4-point shifted PPH subdivision and to the consistent decimation given by Proposition 3 with linear decimation ( 14) is 4.

A numerical estimate of the decay rate of the prediction error is carried out for the shifted Lagrange and the shifted PPH schemes (Figure 1). Two sequences of data are considered : one is a sampling of the function sin(x) in the interval [START_REF] Amat | Analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms[END_REF][START_REF] Deslauriers | Symmetric iterative interpolation processes[END_REF] which is considered as a regular sequence, the other one is a line from image man presented in the left of Figure 2, which is considered as a non-regular sequence. The numerical decay of the prediction error is in full agreement with the theoretical one. For the regular data, the numerical evaluation of the slope is 4.420 for PPH and 5.173 for Lagrange. For the non-regular one, it is close to 1 (0.960 for PPH and 0.917 for Lagrange).

Application to image approximation

In this section, the capability of the two new linear and non-linear multiresolutions for the approximation of real and geometric images (Figures 3 and4) is investigated. Starting from the image of size 512 × 512 (j = 9), several decompositions are performed until j 0 = 5. Then, after truncation of the detail coefficients with different thresholds, the reconstruction transform is applied. Section 7.1 is first devoted to the numerical stability study of the shifted PPH multiresolution transforms. The performance of the two multiresolutions is studied in Section 7.2 and a comparison with existing approaches is given. }. The decomposition stability constant for the l 1 -norm is defined as

C s = ||f j -f j ϵ || 1 ||f j0 -f j0 ϵ || 1 + ∑ j-1 i=j0 ||d i -d i ϵ || 1 .
This constant is evaluated for different columns of the images man (Figure 3) and geometric (Figure 4) with a perturbation generated by a white gaussian noise (N (0, 10)). Figure 2 shows an example of f j constructed from the 260-th column of each image. The numerical estimates of the decomposition stability constant for j = 9 and j 0 = 5 are shown in Table 2 and Table 3 for each image considering shifted PPH and shifted Lagrange schemes. 3. Estimation of the stability constant for the decomposition associated to the shifted PPH and the shifted Lagrange schemes based on image geometric Since the linear shifted Lagrange decomposition is known to be stable, the similarity between the evaluations of the stability constants associated to the two schemes is a good indicator for the stability of the non-linear shifted PPH decomposition.

The same type of evaluation can be performed for the shifted PPH reconstruction and leads to the same conclusion concerning the stability. 7.2. Performance of the multiresolutions. For sake of comparison, multiresolutions constructed from interplatory schemes are also considered. The different multiresolutions are :

• interpolatory Lagrange : The 4-point interpolatory Lagrange subdivision scheme associated to the mask {-1 16 , 0, 9 16 , 1, 9 16 , 0, -1 16 , 0} and the consistent subsampling decimation σ ′ .

• shifted Lagrange : The 4-point shifted Lagrange subdivision scheme given by (3) and the consistent decimation given by ( 14). • interpolatory PPH : The 4-point interpolatory PPH subdivision scheme [START_REF]On a nonlinear subdivision scheme avoiding Gibbs oscillations and converging towards C s functions with s > 1[END_REF] and the consistent subsampling decimation σ ′ . • shifted PPH : The 4-point shifted PPH subdivision scheme given by ( 4)-( 5) and the consistent decimation constructed using Proposition 3 involving the linear decimation [START_REF] Harten | Multiresolution representation of data: A general framework[END_REF].

With the subdivisions and decimations defined above and the associated details subdivisions and decimations given by [START_REF] Deslauriers | Symmetric iterative interpolation processes[END_REF], the construction of the four corresponding compatible multiresolutions is completed.

Reconstructed and original images are compared by computing the so-called PSNR (Peak Signal Noise Ratio) defined as

P SN R = 10 log 10 ( (Maximum Signal Value) 2 Mean Squared Error )
versus the compression ratio defined as the ratio between the size of the original image and the size of the compressed image.

The results on the first image (Figure 3) show that the shifted schemes (that are non-interpolatory) exhibit a better performance than that of interpolatory ones. Moreover, the shifted PPH scheme outperforms the linear shifted one. This difference between shifted Lagrange and shifted PPH schemes is more significant considering a sketchy image such as the geometric one of Figure 4. The result stated by Proposition 3 allows different choices of linear decimation operators and therefore different prediction errors and details. To evaluate the effect of this choice on the performance of the 4-point shifted PPH multiresolution, three linear decimations leading to consistent non-linear decimations are considered in the sequel :

• L8 : decimation hL of length 8 given by ( 14). which is consistent with the 4-point shifted Lagrange and satisfies the contraction of Proposition 3 with constant 15481 20160 . • Quarter : decimation hL given by {- 1 4 , 3 4 , 3 4 , -1 4 } which satisfies the contraction of Proposition 3 with constant 48 64 . Figure 5 displays the evolution of the PSNR with respect to the compression ratio for those multiresolutions. It appears that the choice of the linear decimation has a non negligible effect on the capability of the multiresolution. In this test, the decimation L8, given by ( 14), leads to the best results. This clearly indicates that in practice, the choice of the linear decimation has to be carefully performed before applying Proposition 3 to optimize the performance of the non-linear multiresolution. This point will be studied in future investigations.

Conclusion

In this paper, we proposed a construction of a complete multiresolution framework associated to a general subdivision scheme. A method was first introduced to generate consistent decimation operators, starting from linear subdivision. Then we proposed a generalization to consistent decimations associated to any non-linear subdivision scheme h. It was based on the existence of a linear decimation hL such that hL h -I is contractive. In order to fully characterize the new multiresolutions, compatible detail operators were also constructed. The resulting framework leads to original multi-scale decomposition and reconstruction of data that can benefit from the flexibility of subdivision schemes. Various results have been provided concerning the stability of these operators and the decay of the details. Finally, examples of constructions were developed. They are associated to non-interpolatory schemes for which no multiresolution framework was available before. Their application to image approximation problem illustrated their promising potential compared to classical multiresolution approach.

  A(x, y)) -(H(a, b) -A(a, b))| ≤ max(|x -a|, |y -b|), under the condition (|x| -|y|)(|a| -|b|) < 0 by distinguishing different cases: (1) xy < 0, ab < 0, then H(x, y) = H(a, b) = 0,

Figure 1 .

 1 Figure1. Logarithm of the prediction error norm ||e j || versus the scale j associated to a regular sequence (Top) and a non-regular sequence (Bottom). For comparison, the slopes of the solid red line, the dash-dot black line and the solid black line are respectively 4,5 and 1

7. 1 .

 1 Stability of the 4-point shifted PPH multiresolution. We first provide a numerical evidence of the stability of the decomposition transform. Given a sequence f j = (f j k ) k∈Z , we denote f j ϵ = ((f j ϵ ) k ) k∈Z a perturbed sequence and its associated decomposition {f j0 ϵ , d j0 ϵ , d j0+1 ϵ , . . . , d j-1 ϵ

Figure 2 .

 2 Figure 2. Sequence constructed from the 260-th column of image man (left) and image geometric (right)

Figure 3 .

 3 Figure 3. Top : Test image man, Bottom : PSNR versus compression ratio for interpolatory Lagrange, shifted Lagrange, interpolatory PPH and shifted PPH multiresolutions.

Figure 4 .

 4 Figure 4. Top : Test image geometric, Bottom : PSNR versus compression ratio for shifted Lagrange and shifted PPH multiresolutions.

Figure 5 .

 5 Figure 5. PSNR versus compression ratio for the 4-point shifted PPH subdivision scheme coupled with three different consistent decimation operators, man image

  Definition 6 (Stability of the multiresolution). Let {f j , d j , ..., d j-1 } (resp.

	The reconstruction transform is said to be stable in the norm || • || if there exists a constant C such that for all {f j0 , d j0 , ..., d j-1 } and { f j0 ϵ , d j0 ϵ , ..., d j-1 ϵ } ,
	{ f j ϵ , d j ϵ , ..., d j-1 ϵ	}	)
	stand for the multiresolution decompositions of f j (resp. f j ϵ ).

  

	24367 1152 2975 1152 175 1152 95 1152 175 1152 2975 1152	-63605 1152 -4165 1152 -245 1152 -133 1152 -245 1152 -4165 1152	31115 576 1771 576 875 576 -245 576 -565 576 -10325 576	-10325 576 -565 576 -245 576 875 576 1771 576 31115 576	-4165 1152 -245 1152 -133 1152 -245 1152 -4165 1152 -63605 1152	2975 1152 175 1152 95 1152 175 1152 2975 1152 24367 1152

Table 2 .

 2 Estimation of the stability constant for the decomposition associated to the shifted PPH and the shifted Lagrange schemes based on image man

	column index	50	120	190	260	330	400	470
	C s -Lagrange 1.3256 1.3646 1.3420 1.3456 1.2602 1.3248 1.3647
	C s -PPH	1.2337 1.2668 1.2307 1.1794 1.2353 1.2920 1.2679

  • L12 : decimation hL of length 12 given by

	{	19 16128	, -	19 11520	,	19 576	, -	19 576	, -	2623 16128	,	7639 11520	,	7639 11520	, -	2623 16128	, -	19 576	,	19 576	, -	19 11520	,	19 16128	}