Mircea Moca
email: mircea.moca@econ.ubbcluj.ro

Darie Moldovan
email: darie.moldovan@econ.ubbcluj.ro

Oleg Lodygensky
email: oleg.lodygensky@lal.in2p3.fr

Gilles Fedak
email: gilles.fedak@inria.fr

E-Fast & CloudPower: Towards High Performance Technical Analysis for Small Investors

Keywords: E-Fast, CloudPower, Technical analysis, Moving averages, High Performance Computing

About 80% of the financial market investors fail, the main reason for this being their poor investment decisions. Without advanced financial analysis tools and the knowledge to interpret the analysis, the investors can easily make irrational investment decisions. Moreover, investors are challenged by the dynamism of the market and a relatively large number of indicators that must be computed. In this paper we propose E-Fast, an innovative approach for on-line technical analysis for helping small investors to obtain a greater e ciency on the market by increasing their knowledge. The E-Fast technical analysis platform prototype relies on High Performance Computing (HPC), allowing to rapidly develop and extensively validate the most sophisticated finance analysis algorithms. In this work, we aim at demonstrating that the E-Fast implementation, based on the CloudPower HPC infrastructure, is able to provide small investors a realistic, low-cost and secure service that would otherwise be available only to the large financial institutions. We describe the architecture of our system and provide design insights. We present the results obtained with a real service implementation based on the Exponential Moving Average computational method, using CloudPower and Grid5000 for the computations' acceleration. We also elaborate a set of interesting challenges emerging from this work, as next steps towards high performance technical analysis for small investors.

Introduction

The nowadays financial markets are amongst the most competitive markets in the world. More, the specific conditions of the financial markets are rapidly changing, so that e cient data analysis, fast and well grounded reactions to market changes, multiple-source market data integration are actual challenges concerning investors. Obviously, the big players of the market like the large capitalized banks a↵ord to address these challenges by developing and maintaining sophisticated systems relying on modern HPC infrastructures that provide advanced financial analysis capacities. More, they support research departments with specialists that are inter-disciplinary skilled in areas like maths, finance and programming. In contrast, for smaller players that do not have access to the above described resources it is almost impossible to e ciently perform in these market conditions. The poor investment decisions lead to about 80% of the financial market investors to fail [START_REF] Barber | Chapter 22 -the behavior of individual investors[END_REF].

In this paper we propose E-Fast, an innovative platform prototype for online technical analysis for helping small investors to obtain a greater e ciency on the market by increasing their knowledge. The prototype relies on High Performance Computing (HPC), allowing to rapidly develop and extensively validate sophisticated finance analysis algorithms. The novelty stands in the initiative of building a real technical analysis solution based on HPC for helping small investors to cope with hash market challenges. Our approach is designed with the potential of being sustainable through the specific communities' e↵orts. For instance, we use technologies (like XtremWeb middleware) that allow an easy aggregation of computing resources from the participating individuals.

In this work, we aim at demonstrating that the E-Fast implementation, based on the CloudPower HPC infrastructure, is able to provide small investors a lowcost and meaningful service that would otherwise be available only to the largest finance institutions. We describe the architecture of our system and provide design insights. We also present the results obtained with a real implementation of the Exponential Moving Average computational method, using CloudPower and Grid5000 for the computations' acceleration. We also elaborate a set of interesting challenges emerging from this work and explain our envisioned approach.

Our e↵orts also come to meet a set of main objectives of the European strategy for the 2014/2020 timeframe [START_REF]europe 2020 -a strategy for smart, sustainable and inclusive growth[END_REF], focusing on the following three priorities:

-Intelligent and sustainable growth by developing a knowledge and initiative based economy.

-Sustain a competitive and e cient economy that relies on its own resources.

-Social and territorial cohesion support. In this context, direct extensions of our approach can support the building of small investor communities.

The remainder of this paper is organised as follows. Section 2 gives the background of our work, section 3 presents the architecture of the implemented E-Fast prototype, while section 4 presents the experiments run with the prototype and discusses results. Further, section 5 presents related work, section 6 explains relevant future challenges and section 7 concludes this work.

In this section we describe the context of our work and define the concepts used in our discussion. In our approach, the user interacts with the system through a set of services. Such a service is characterized by: -Service name, -Input structure, -Result (output) structure, and -Other specific parameters. The user aims at executing one or more services in order to analyze their specific results. The results can either be directly interpreted by the user or further used into third party analysis tools. The logic of a particular service is given by a stand alone or a mix of computational method(s). Examples of methods may be moving averages, genetic algorithms, data-mining algorithms, Monte Carlo simulations and others. For each service, the system expects to a strict definition regarding the input and output data.

A service execution is the process of executing the computational method(s) underlying that service on particular input and other user-defined parameters. An execution yields a result (output) with specific semantic and format. The execution is managed by the E-Fast server and may imply the delegation of the e↵ective computational method execution to the HPC infrastructure (see system architecture in section 3.1). The general execution flow for a particular service is detailed in section 3.2.

Moving Averages

In this work we built a service that is simply defined by the Exponential Moving Average computational method. Forward we give a short description of this method and explain its use in technical analysis.

The employment of moving averages in technical analysis is already a common approach. Their simplicity and e ciency made them successful in industry and economics even before being applied in the financial area. By definition, the moving average (denoted by MA(k)) is a method to smooth the variability in time of the analyzed time series by considering the average of the most recent k observations. In technical analysis, to provide a trading signal, two moving averages for the prices of a certain stock are calculated. One of them has a larger number of observations k l , in order to capture the long term trend and the other with a smaller k s , which is more sensitive to the recent fluctuations in price. When the value of the short term moving average passes above the long term moving average, we have a buy signal, suggesting that the price will increase more. On the contrary, when the short term moving average crosses the long term moving average by dropping below its value, it signals sell, forecasting a drop in the price of the stock.

There are several versions of the method, like simple (SMA) and exponential (EMA) [START_REF] Holt | Forecasting seasonals and trends by exponentially weighted moving averages[END_REF][START_REF] Cox | Prediction by exponentially weighted moving averages and related methods[END_REF] moving averages. The simple moving average can be criticized because gives the same importance to all considered observations. In contrast, the exponential moving average gives a higher importance to the recent observations, gradually decreasing the importance for the older ones.

The calculation of EMA is given in eq. 1, where EM A(k) is the exponential moving average calculated for k periods, a is the smoothing constant, showing the importance weight to give to each component and P is the price of the stock at a certain moment.

EM A(k) = ↵P k + ↵(1 ↵)P k 1 + ↵(1 ↵) 2 P k 2 + ... + ↵(1 ↵) k 1 P 1 (1)
The moving averages can be used in technical analysis to calculate trading signals (buy or sell). For this, the moving average must be calculated twice, with di↵erent values for k. We call long term EMA (EM A l ) the one calculated on a bigger k and short term EMA (EM A s ) the other one. For a better understanding of using EMA in this context, in the following we provide an example, depicted in figure 1. For a particular stock, we show the calculated moving averages and the buy/sell signals, also marking the trading points. The chart contains data from March 2012 to March 2013 on the stock price of Microsoft, where k = 12 for EM A s and k = 50 for EM A l . We marked every intersection between the lines of the EMAs with a dot line. Depending on the stock's behavior on the market, this method can yield a large number of buy/sell signals. In practice, not all of the generated signals are worthy to be considered. Hence, in order to avoid noisy trading signals, generated by two or more very close in time crossovers between the two lines, we used a supplementary filter (represented by the dash line). Only when EM A s passes the filter, a trade is generated. We also mention that our strategy permits both long and short positions and we do not discuss the profitability of this particular example.

Going into detail, we notice that the first signal (in May 2012) is a sell signal. The transaction Sell Trade (1) only occurs after the value of the EM A s dropped below the filter value. After this transaction, we own a short position. Next we have the Sell Signal (2), which does not lead to to a trade since EM A s doesn't surpasses the filter. Forward, another sell signal is generated, but even if the value of EM A s went bellow the filter value, the signal is ignored since we already own a short position (and we wait for a buy signal). The following signal, Buy Signal (2) flags the opportunity to buy, which occurs at the moment showed by Buy Trade (1) arrow, in this way closing the short position. Sell Signal(3) is shortly followed by Sell Trade (2), opening a new short position, which is closed only several months later, at Buy Trade [START_REF]europe 2020 -a strategy for smart, sustainable and inclusive growth[END_REF]. Although the strategy is quite simple, finding e cient values for its parameters (like k, the smoothing factor ↵ or the size of the noise filter) can be di cult. In fact, they are very sensitive to the trends inside the analyzed time series, making it di cult for the investor to find the best solution. For this reason, an almost exhaustive search for the best parameter combinations for a strategy that works usually only for a particular stock can be very costly in terms of computational time.

CloudPower

Big Data and High Performance Computing (HPC) are key factors in knowledge and innovation in many fields of industry and service, with high economic and social issues: aerospace, finance and business intelligence, energy and environment, chemicals and materials, medicine and biology, digital art and games, Web and social networks.

Today, acquiring high-end data centers and supercomputers is very expensive, making Big Data and HPC unreachable to small business for their research and development. For reasons both technical and economic, access to such technologies is di cult to fundamental actors of growth that are small and medium-sized innovative companies. That is why it is important to support them in this process. This a fundamental tool for competitiveness and innovation capacity of service and industry enterprises.

The CloudPower project o↵ers a low cost Cloud HPC/BigData service for small and medium-sized innovative companies. With CloudPower, companies and scientists will run their simulations to design and develop new products on a powerful, scalable, economical, reliable and secure infrastructure. CloudPower leverages on the open-source software XtremWeb-HEP previously developed by the CNRS and INRIA in France. The principle of the technology is to collect the under-exploited resources on the Internet and Data Centers to build a virtual supercomputer providing HPC and Big Data services on demand. CloudPower is supported by the French National Research Agency (ANR).

The E-Fast prototype

In this section we give a detailed description of the proposed E-Fast prototype, focusing on its architecture as well as the service definition and execution.

System architecture

In this subsection we present the architecture of the implemented E-Fast prototype. This mainly contains three components: the E-Fast client and server and the distributed computing infrastructure.

Figure 2 depicts the architecture of the prototype. First, the E-Fast client runs on the users' machines and allows them to connect to the E-Fast server and consume its services. This is a web interface that allows the client to set up the parameters of the execution and to inspect and analyze the results obtained from the executions. Based on the execution parameters received from client, the server extracts the appropriate input data from the database and transfers it to the distributed computing infrastructure. Then, it delegates to the infrastructure the execution of the service on the respective input data. After receiving the results from the infrastructure, the server composes the final result from a user's perspective and delivers it through the client. The database managed by the server mainly contains: financial market historical data, results of the executions and execution meta data.

Service execution flow

In this subsection we detail the overall approach for a service execution.

Figure 3 presents the general execution flow for a particular service. The process begins with the user, specifying the execution parameters through the E-Fast client. Based on these parameters the E-Fast server makes the decision whether to start a new execution or directly deliver the results corresponding to the received parameters. The direct delivery would occur when the user has already performed an execution for the same parameters. If it's not this case, the E-Fast server selects from the database the appropriate input data (according to the computational method defining the required service and the user's preferences). Forward, according to the computation distribution strategy of the service, the server creates a set of input files and uploads them to CloudPower. Then, the server starts the execution of the service by creating and launching a job on the DCI for each input file. After a while, the E-Fast server receives the results from CloudPower, aggregates them into a final result and presents them to the user within the E-Fast client.

Computational method distribution

As discussed in section 1, the computational methods used in technical analysis are generally computational intensive. In this sense, our implementation relies on CloudPower for accelerating the service execution. This is capable of executing bag of tasks, these being independent one from another. In this context, for any given computational method, this aspect brings the challenge of defining a computation distribution strategy. This is significantly determined by the method's definition, so when a high execution performance (i.e. in terms of makespan) of the service must be attained, then the distribution strategy may be critical. In this case, it must be tailored to the method's definition. In the following we call distribution criterion a concept based on which a system designer defines the distribution strategy.

Since the Moving Averages are computed on a single stock at a time and the obtained results are independent from those of other stocks, we chose the stock to be the distribution criterion. Hence, our implementation executes the computational method on time series, separately for each stock. For example, if the user wants to calculate the rentability of the S&P 500 index, then the system runs a task for each stock from the whole set of 502 stocks (this is our experimental data and it is described in section 4).

For each execution, the user specifies a time series from the available historical data. The real execution time of the service on a specific machine from the HPC infrastructure depends on the size of the specified time series. From this perspective, if one aims at optimizing the execution, then the distribution strategy can be changed -i.e. execute a task on a set of stocks, not only a single one. By this, a system designer could adjust the average real execution time of a task in order to fit the HPC infrastructure's requirements in terms of optimality.

Service architecture

In our approach, the functionality of a service is organized in two components: the technical and the business layers. While the technical layer implements one or more computational methods, the business layer computes business keyvalues (like profit), with direct utility for the financial analyst, based on the technical layer's output. Figure 4 depicts the service layers and parameter types. For a better understanding, we explain the parameter types that characterize the service directly referring to the Moving Averages method. Hence, the layers contain the following parameter types: -P 1 is a parameter type that directly a↵ects the execution of the service on E-Fast client side. For instance, the user may vary the commission value applied to the gross profit. These computations have low complexity and can be directly supported by the E-Fast client and the input data is directly available on the E-Fast server.

-P 2 has direct impact on business but the implied computations are more complex compared to those of P 1 , so they need to be performed by the E-Fast server with the eventual delegation to the HPC infrastructure. Such a parameter would be the filter used to compute the e↵ective transactions based on the signals provided by the Moving Averages method. The adjustment of the filter may lead to a significantly di↵erent number of transactions, making the service more sensitive to the stock's volatility. Consequently, more transactions lead to a greater absolute value of the calculated trading fees. Hence, the impact of this parameter is important.

-P 3 is specific to the technical layer and is related to the optimization of the computational method and generally require the E-Fast server to coordinate the execution on the HPC infrastructure. For instance, finding optimal k s and k l values for the Moving Average method and particular market data implies the a costly execution of the service for a relevant set of k s and k l combinations on the HPC infrastructure.

Experiments and results

The experimental data used in this work was historical market data of the companies composing the S&P500 index for the 2006-2014 period. This is one of the major indices in the United States, covering around 75% of the equity market by capitalization. It comprises 502 stocks for which we collected daily pricing information.

In order to obtain the best parameters for the Exponential Moving Average trading strategy, one execution of the E-Fast service computes for one stock from S&P500 the return based on EM A for all combinations of k. More specifically, k s = {2, 3, 4 . . . 50} and k l = {4, 6, 8, . . . 100}. This setup led us to a number of 496 603 profitability calculations for the whole S&P500 index. In our experiments we used the same noise filter value, but in a real system, the user must be able to tune this parameter.

Our main success criterion was the profit obtained by each combination. When calculating the profit we used the same percentage fees per trade to be extracted from the results of each sell-buy pair. Other criteria that can be used for evaluating the performance of a trading strategy can be used, depending on the investment objectives. Such criteria can be the maximum drawdown (the maximum loss of a strategy during its life), number of profitable trades / all trades, risk adjusted return, Sharpe ratio etc.

We compared the results obtained with our trading strategies against the buy and hold strategy on the S&P500 index. While the total return for the S&P500 for the entire analyzed period was a profit of 61.84%, the results of our strategies show a profit of 77.3%. Table 1 shows the returns obtained by the best 5 groups of strategies. We observe that there are several k s and k l combinations yielding significant returns, the di↵erences among them being relevant. This demonstrates the importance of finding and using the right parameters for a certain strategy. In figure 5 we show the distribution of the obtained returns. We note that all strategies obtained positive returns, some of them with outstanding positive performance. These performances can be filtered and used as main target investments by the investors. Although they are shown as outliers, they are actually the most important strategies, the ones that provide the highest returns. For example, the best strategy for the Apple stock (AAPL) is obtained using the short term moving average with k s = 14, long term moving average with k l = 52 and a filter of 0.15%. While the main focus in our work is to show how E-Fast & CloudPower can be used together for solving computational intensive financial problems, we did not focus on refining and optimizing the business results.

In practice, the methods can be optimized by periodically running the service (e.g. weekly), so that it adapts to new, recent market data. Then, the user may retain the best parameter combination till next analysis.

The literature in the field of distributed and parallel computing in finance is not very rich in terms of diversity. Most of the related work is focused on solving particular computational finance problems involving Monte Carlo models for derivative pricing. Since the Black and Scholes [START_REF] Black | The pricing of options and corporate liabilities[END_REF] model was introduced, it was widely used, along with others, to compute derivatives prices. The iterative character of the method (usually it can reach 1 million runs) can be very consuming in terms of time and computation resources. This can be unpractical in the field of finance, where the result is expected in terms of minutes, seconds or close to real time.

Stokes-Res [START_REF] Stokes-Ress | Managing parallel and distributed monte carlo simulations for computational finance in a grid environment[END_REF] discusses the parallelization and distribution of Monte Carlo option pricing in the context of several grid environments. Di culties related to running parallel tasks on the grid were revealed, consisting in synchronization of the start task time and partitioning the computational load. They conclude that several other tests should be carried on this and other computational finance problems in order to have a better image on the performance that can be achieved. Heinecke [START_REF] Heinecke | Many-core architectures boost the pricing of basket options on adaptive sparse grids[END_REF] presents a scalable approach for the Black and Scholes solver for pricing options. They discuss the results obtained in the context of several hardware solutions, emphasizing the robustness of their proposed operator for addressing this particular issue.

In the context of finding the best trading rules by using technical analysis methods, Strassburg [START_REF] Heinecke | Many-core architectures boost the pricing of basket options on adaptive sparse grids[END_REF] employs parallel genetic algorithms. Due to the data dimension, the repetitive characteristics of the algorithm and time-critical answer need, the high computational resources are a must for the institutions and individuals dealing with sophisticated trading decision support methods. The paper presents the e↵orts to optimize the parameters of the technical indicators used in order to improve the trading results. The tests were conducted on the Madrid Stock Exchange Index data, but only on holding long positions (not allowing short selling) aiming to demonstrate the usefulness of the parallel approach. However, the tests were made on a single machine with four cores, limiting the conclusions to this closed environment. A solution to reduce the computation time for intensive operations in financial analysis was proposed by Moreno [START_REF] Moreno | Speeding up large-scale financial recomputation with memoization[END_REF]. They built a disk memoization solution to reduce the number of repeated computations, allowing the analysts to build and debug their algorithms more quickly.

A new automated trading system architecture was introduced by Freitas [9], who considers dividing the trading problem into several tasks handled by distributed autonomous agents, with minimal central coordination. Their strategy is based on obtaining a consensus trading signal based on other trading signals from multiple strategies. The results were considered very satisfactory after testing moving average crossover strategies on a large database.

The work so far consisted in building the E-Fast prototype equipped with a first service based on the Exponential Moving Averages computational method. From both -technical and financial perspectives, the aim was to proof the concept of providing advanced technical analysis services based on distributed computing, for small investors. These achievements led us to a set of interesting challenges and motivates us to build further on this concept. The most relevant development directions on this topic are:

-Building service composition mechanisms that allow the user to combine several services in order to obtain advanced knowledge on the market. The interesting aspect of the services in this context is that based on the same market data, several computational methods can be applied in order to obtain similar or complementary visions and understanding of the market's behavior. For instance, the user can execute two services in order to obtain results with similar semantic but computed with di↵erent methods. In this case, the aim would be to compare the di↵erent results obtained for particular input data and assess a degree of confidence in them. In other situations, the user might be interested in combining two or more services by running them and using their outputs together in order to understand the market from di↵erent perspectives.

-Building advanced collaboration mechanisms to allow the di↵erent stakeholders work together and obtain synergy. As mentioned, players on the financial market need high expertise in fields like mathematics, finance and technical/computer science. In this sense we aim at building mechanisms that allow users to share their resources (i.e. computing, knowledge, expertise, skills). By this we aim at facilitating the crowding of mutually interested specialists, small investors and resource owners.

-Optimizing the prototype from two perspectives. For each service, from a technical viewpoint, the system designer should optimally decide the computations performed within the two technical and business layers of the service (presented in section 3.4). This is very challenging, since the design decision is specific to each computational method. More, the solution must take into account the specifics of the HPC infrastructure and, at the same time be parametric, in order to allow the user to choose amongst several service flavours. From a financial perspective, the system must keep services up-to-date from a business viewpoint. As explained in section 4, optimal key-parameter values must be found for a computational method in order to e ciently adapt to the recent market behavior. In this sense, the challenge would be to build fully or quasi-automatic mechanisms that keep the services up-to-date and adapt to market fluctuations.

In this paper, we proposed E-FAST, a service prototype for on-line technical analysis that can support small investors to obtain a greater e ciency on the market by increasing their knowledge. In this work we created a prototype that relies on High Performance Computing (HPC). This allows one to rapidly develop and extensively validate sophisticated finance analysis algorithms. We aimed at demonstrating that E-Fast based on the CloudPower HPC infrastructure is able to provide small investors a scalable, low-cost and secure service that would otherwise be available only to the largest financial institutions. We presented the architecture of our system. We also presented the results obtained with a real implementation of the Exponential Moving Average computational method, using CloudPower and Grid5000 for the computations' acceleration. Finally we presented a set of interesting challenges emerging from this work and describe our approach to address them.

Fig. 1 .

 1 Fig. 1. Stock price, and computed moving averages and signals.

Fig. 2 .

 2 Fig. 2. Overview of the E-Fast prototype architecture.

Fig. 3 .

 3 Fig. 3. General service execution flow.

Fig. 4 .

 4 Fig. 4. Service general architecture.

Strategy

Fig. 5 .

 5 Fig. 5. Returns distribution for the S&P500 index.

Table 1 .

 1 Return for top 5 groups of strategies.

	Strategy group	1	2	3	4	5
	Return	77.3% 73.7% 71.3% 69.6% 68.5%


Aldo Fortunati

Research/action projects on childhood education by sharing experiences in the framework of the EU for the development of quality ECEC services

Aldo Fortunati and Arianna Pucci Istituto degli Innocenti, Florence, Italy Good morning everyone, I'm pleased to be here and I'd like to give my thanks to the organizers for the opportunity to present our experience in this group. I'm ALDO FORTUNATI and I'm the Head of the educational department of the Istituto degli Innocenti of Florence. I'm here with ARIANNA PUCCI, who is one of the researchers of my staff. She has worked with me for the last two years on the TALE project.

This presentation will be divided into three parts:

1. the first part will present the principal elements of the so-called Tuscan Approach to children's education;

2. the second part will be about the TALE project;

3. and the third part serves to give you some final considerations about results and perspectives for the future work.

We are pleased to have with us today a complete documentation of the experience that we are going to present: the book "TALE: Tuscan Approach learning for early childhood education and care. Activities, results and perspectives" including a video, recounting the activities we have developed in the last two years. We hope that their dissemination may attract useful comments and observations. This does not exonerate us, but may help us, with this talk, to present the work we have done and talk of the possible support that this work can give to support similar reflections in other contexts.

Our point of departure is the valuable experience developed over the last thirty years in Tuscany, one of the Italian regions in which policies and experiences involving childhood educational services have been developed in a privileged context. Our first difficulty was that of defining this experience, because when we speak of a positive experience, we tend to embellish and transform it into a model, as if it could be easily packaged and replicated, irrespective of the context where it is produced. We do not intend to commit such a mistake talking about the Tuscan Approach to child education. Moreover, it would mislead us, for the Tuscan approach has always been built upon exchanges of similar experiences and comparisons of diversities. Therefore, diversities are not faults, rather basic elements useful also in dynamising the Tuscan approach itself, and opening it to innovation and change.

It should be clear that pedagogy is not made of abstractions. It relies on guidelines to be put in a context, giving value to resources and potential coming from the communities where the services are in place.

Therefore, the solution was to try to identify key concepts and find a balance between them.

The first three words we identified were: politics, pedagogy and experiences.

In this context, the Tuscan Approach to children's education is interestingwe thinkbecause it demonstrates that three of the following elementswhich are not always linkedcan be well balanced:

 a clear and durable policy commitment in the field of child education;

 updated and accurate pedagogical research;

 an active and constructive involvement of the local resources in developing the experiences.

But the word "approach"was the one we chose to assert an identity not blocked within the bounds of a model. Starting from the beginning, there are five points that we selected to describe the peculiarity of the Tuscan Approach to children's education and they are:

Governance

Space design

When we speak of space design and education, we refer to the importance of spacearchitecture, furnishings and materials -to support children's experiences and also children's processes of growth, social relationship and learning. The idea of a space constructively supporting the children's experience belongs to the most recent pedagogical reflections, as long as these prove sensitive to offering the child stimuli and opportunities already within the education context and prior to the initiative of the single educator.

Flexible curriculum

The investment in qualifying space helps us build an idea of a flexible curriculum, open and above all oriented towards a shift from attention to contents to attention to the conditions of the experience offered to children. When speaking of child education, we do not need to prepare the contents table, but we need to prepare the table of conditions, the opportunities offered for children to experience. Investing in space as a resource helps us to develop an indirect pedagogical perspective in which the adults are able to make proposals not only through their direct action but also and above all through the potential that the framework of opportunities that they have organised offers for the children to experience.

Participation and family education

Although in the Tuscan Approach the central role is given to the children, at the same time nobody should underestimate the crucial involvement of families in the development of the education project. What we have in mind is the importance of considering families as real, active actors in the development of the educational project for children. Moreover, the involvement of the families in the education service is also enriched by the numerous occasions of spontaneous meetings and exchanges, which can themselves be considered a sort of familyeducation context.

Staff qualification and training

In relation to the qualification and training of the staff, we think that in this point it is very important to stress the connection between basic and in-service training, understanding that this kind of connection means also to consider training as a research/action process that means intentions, aims but also opportunities to reflect together, in the group of educators, in order to develop innovation and new projects.

Governance

When we talk about governance, we have in mind not only to develop rules and laws to control ECEC services. We are thinking of a way to control the processes for developing quality, working close to the services, in the field, sharing aims, but also reflections, with educators and families.

A children's school needs proximity and not distances, nor high-sounding designs, but to be built around an activism able to valorise resources and potential that are found above all within the communities. The Tuscan experience has certainly represented this too andas has been pointed outwith the municipalities' extremely important role and the great skill employed in putting public initiatives in synergy and not in contrast with private initiatives.

These are the fundamental elements of the Tuscan Approach, which doesn't correspond to a modelas we said beforebut instead to an approach, which means a way to consider and valorise diversities.

We must remind ourselves of the central theme of keeping a balance at this point, because the value of the 5 points I have just listed, lies in the ability to include them in a systemic vision of the project and to keep them together. These elements have thus formed the basis not only for recounting our experience to others, as well as to ourselves, but also for building a dialogue and discussion around them, aimed at building innovative paths of experimentation.

This is what we had in mind when we started thinking of the TALE project, founded by the European Union and involving IDI, UNIFI and some other organizations from Bulgaria, Lithuania and Slovenia.

TALE is an acronym and it means Tuscan Approach Learning for Early childhood education and care. We think that training is an important activity to support professional development, in the process of permanent learning. This calls for integration of work practices and theoretical reflection within a framework of exchange and comparison of different experiences.

The TALE project is precisely this: it puts the traditional research-action approach together with an exchange of experiences, set in a European trans-national context.

In this sense, the Tuscan Approach should not be seen as a model or a good practice to be repeated in other contexts but rather, as an opportunity to reflect on some important "markers" of quality in the early childhood education services, as Aldo Fortunati has already mentioned.

We can divide the project into 3 key steps:

1. workshops;

training module in Tuscany;

3. projects for innovation and experimentation.

At first three workshops were organised: in Vratsa, Ljubljana and Vilnius. After selecting an inspiring case study, the Tuscan Approach, a number of opportunities and learning settings are provided where the phenomenon is observed, split into its basic dimensions, proposed for comparative analysis, used as a term of reference to start a process of quality improvement involving individuals, organizations and systems. During the workshops abroad, we presented the contents of the Tuscan Approach, also showing a video that we have made; we dedicated a long time to listen to the presentation of the local educational system and then we visited some local ECEC services. So these occasions were useful for knowing and making known.

After that a training module in Tuscany was organised, where educators, managers and pedagogues from the 3 countries were gathered together. They met with Tuscan experts, professionals and educators and spent some time in several Tuscan ECEC services. The training module took one week. After a general introduction, specific sessions were dedicated to 5 fundamental elements. Governance and training were dealt with in seminars organized by the Istituto degli Innocenti and the University of Florence. The pedagogical issues were addressed with the involvement of some of the most representative Tuscan Municipalities:

 Pistoia raised the issue of space design;

 San Miniato tackled the curriculum;

 Livorno spoke about family education and participation.

It's important to stress that the training module combined the theory with the practice because the 3 country groups spent the larger part of their time visiting Tuscan services and meeting children, parents and educators at work.

After these two steps, each partner country's team was able to discuss its own plan of action for some innovations and experimentation to be undertaken in the following year. Also in this case as I have already statedthe fact that every team was comprised of different professionals, such as managers, pedagogists and educators has stimulated the group to discuss the various points of view and we believe this is an important aspect of our training experience. These innovation projects were continued during a school year and the country groups presented the first follow-up at an International conference in Florence at the end of the project. The final results are original and different and the future prospects too, but our capacity to reflect on the quality of the experiences is growing and going deeper.

Communication and intense exchange of data sustained the process of analysis and action, with rich documentation following all steps. We also paid particular attention to visual communication, and we created a web site to share videos, photos and other materials...This volume, including the video the Tuscan Approach to children's education, an experience open to dialogue, it is also an important tool of documentation.

In conclusion we think that the documentation of such research/action initiatives may contribute to actively supporting the coordination and development of the ECEC services and polices in Europe, in accordance with the orientation towards mutual exchange.

Reflecting on the prospects that may open up, based on the experience and activities completed in the TALE project, firstly I want to refer to the fact that in the work we have shared together in the past two years, we have tried to speak of child education.

We have put together two expressionsidentity and diversityseeking to understand how these two terms can enter into a dialogue and in this dialogue, how it can be possible to conceive, represent and practice an idea of education that is full of important potential.

I also want to stress that in a Europe that does not represent itself as very capable of holding identity and diversity togethernor of welcoming different identities -I believe it is very interesting and positive to acknowledge that reflecting about child education today may provide a fertile terrain for meeting, exchanging and sharing thoughts to increase our capacity to plan and build the future through this means.

In this case too, because the European community has enormous quantitative and qualitative differences in relation to ECEC services, we should also consider the fact that it is not so easy nor perhaps usefulto quickly establish general principles that are valid for all.

This does not make us underestimate the importance of the guidelines and reflections on quality that have been produced from the 1990s until the present day in Europe, but it makes us think that it is not easy to create a mechanism for implementing quality based on guidelines coming from above.

We do think that this idea of developing experiences, starting from a dimension where there is a more horizontal discussion, is really very useful and important.

We hope that the TALE experience will also help us to confirm that comparison with the experience of othersfor the adults as well as the childrenis very important for helping us better understand what we are doing and also what we can do more and better.

For any information and to receive a free copy of the publication "TALE: Tuscan Approach learning for early childhood education and care. Activities, results and perspectives", please contact:

fortunati@istitutodeglinnocenti.it pucci@istitutodeglinnocenti.it

To download the video included in the book link the following Url:

goo.gl/5CaG6z


Cédric Gérot

Sylvain Meignen

Multiresolution Framework on the Interval based on Fibonacci Tiling, B-splines and Lifting Scheme

In this paper, we introduce a multiresolution analysis on the interval based on non-uniform Bsplines defined on the Fibonacci tiling. The construction of the multiscale structure based on the substitution rules of an L-system allows the derivation of a known framework from the regular dyadic setting to a non-uniform setting while limiting the number of different filters to a few and keeping a similar stability. After having explained how our approach fits into a biorthogonal framework, we detail how to build analysis wavelet functions in the B-spline setting. Then the emphasis is put on the definition of boundary scaling and wavelet functions by means of scaling equations. Our implementation of the multiresolution structure is done in such a way that the computation is carried out in place. Finally, a numerical analysis of the stability of the proposed scheme shows its similar behavior to the same multiresolution analysis that would be derived on a dyadic sampling.

Introduction

Wavelets built on an irregular grid hierarchy are not translates and dilates of a single mother wavelet and lead to multiresolution transforms with, in the most general case, as many different rules as there are samples [START_REF] Daubechies | Wavelets on irregular point sets[END_REF]. However it is possible to define an irregular grid hierarchy with a quasicrystal structure which allows us to define wavelets from a finite number of mother wavelets [Bernuau, 1998]. Andrle et al. use substitution rules to define such a quasicrystal grid hierarchy and the associated semiorthogonal B-spline wavelets on the line with the aim to do a wavelet analysis of quasicrystal diffraction patterns [START_REF] Andrle | Bernuau spline wavelets and sturmian sequences[END_REF]. With zero-degree B-spline basis functions, a full multiresolution framework can be implemented, but with higher degree basis functions, of the reconstruction and decomposition sides of the associated multiresolution framework, only the former can be efficiently implemented. Indeed, as explained by Lyche and Quak for their construction of B-spline wavelets on the interval, semiorthogonality makes the design of a good decomposition algorithm difficult [START_REF] Lyche | Theory and algorithms for nonuniform spline wavelets[END_REF], Quak, 2002]. Since a direct computation of the inverse of the reconstruction leads to full matrices, the decomposition is rather applied by solving a linear system which can be different at each resolution. On the contrary, we want to decompose the signal using a finite number of localized filters. To do so, we drop semiorthogonality for general biorthogonality, which allows us to design non-uniform B-spline wavelets on a hierarchy of irregular grids defined by the substitution rules of Fibonacci L-system [START_REF] Nivoliers | L-system specification of knot-insertion rules for non-uniform B-spline subdivision[END_REF], and associated with local linear decomposition and reconstruction making up a perfect reconstruction filter bank. Our construction, inspired by the designing of second generation wavelets with a lifting scheme [Sweldens, 1998], is made up of four successive steps presented in successive sections in this paper. In §2, we define a multiresolution analysis, i.e. a set of nested spaces spanned by scaling functions. In §3, we build a subdivision scheme from refinement relations between scaling functions. In §4, we carry out in-place implementation of the perfect reconstruction filter bank based on the previously defined subdivision scheme. In §5, we stabilize the filter bank by means of extra updating elementary steps. Finally, since we aim to deal with data displayed on an interval, specific subdivision rules and update filters are defined at the interval boundaries in §6. Our construction can be viewed as the derivation of biorthogonal B-spline wavelets defined in the regular dyadic setting by [START_REF] Cohen | Biorthogonal bases of compactly supported wavelets[END_REF] and their adaptations to the interval [Bittner, 2006, Primbs, 2010, Černá and Finěk, 2011] to a Fibonacci tiling on the interval, while keeping similar stability properties as shown by numerical experiments presented in §7.

Multiresolution Analysis

Let S be a space of functions defined on a 1D-domain Ω. A multiresolution analysis of S is a sequence of closed subspaces {V j ⊂ S | j ∈ J ⊂ N} so that [Sweldens, 1998]:

1. V j ⊂ V j+1 , 2. j∈J V j is dense in S,
3. for each j ∈ J , V j has a Riesz basis given by scaling functions {ϕ j,κ | κ ∈ K(j)}, where K(j) is an index set satisfying K(j) ⊂ K(j+1). When successive elements of K(j) will be considered, we will name them {t j,k | k = 1, . . . , #K(j)}.

In the first generation wavelet setting, scaling functions are translates and dilates of one mother scaling function. On the contrary, the second generation wavelet setting allows to deal with hierarchies of irregular grids without mother scaling function [START_REF] Daubechies | Wavelets on irregular point sets[END_REF]. Our goal is to define a setting which is in-between, based on a hierarchy of irregular grids and where scaling functions are defined as translates and dilates of a small set of different functions. We define the scaling functions ϕ j,κ as non-uniform B-spline functions of order d+1 (piecewise polynomial of degree d) defined on d + 1 successive intervals [t j,k , t j,k+1 ], . . . , [t j,k+d , t j,k+d+1 ] where κ = t j,k , and normalized in the sense given in [de Boor, 1976]:

ϕ j,κ (x) := t j,k+d+1 -t j,k d + 1 M t j,k ,...,t j,k+d+1 (x),
where M t j,k ,...,t j,k+d+1 (x) = (d + 1) [t j,k , . . . t j,k+d+1 ] (•x) d + is the B-spline of Curry and Schoenberg, x d + = (max {0, x}) d and [t j,k , . . . t j,k+d+1 ] f denotes the (d + 1)-th divided difference for the function f at the points {t j,l } k+d+1 l=k . In [de Boor, 1973], it is shown that {ϕ j,κ | κ ∈ K(j)}, is a Riesz basis in L ∞ of V j . Note that if we were interested in other norms L p , 1 ≤ p ≤ ∞, we could choose

((d + 1)/(t j,k+d+1 -t j,k )) 1/p ϕ j,κ as a Riesz basis in L p of V j .
A sufficient condition for these scaling functions ϕ j,κ to be defined as translates and dilates of a small set of functions is to impose any interval [t j,l , t j,l+1 ] to have its length in a set {a

(j) i = a i /ρ j , i ∈ I ⊂ N },
where ρ is a constant ratio and I a finite set. As shown in [START_REF] Nivoliers | L-system specification of knot-insertion rules for non-uniform B-spline subdivision[END_REF], an L-system is a convenient tool to define such a set of intervals with a finite set of rules describing how to split each interval of length a (j) i into intervals of length a (j+1) i . An L-system is defined by a set of symbols {A i , i ∈ I}, a set of rewriting rules

A i → A i 1 . . . A ir , (2.1)
where (i 1 , . . . , i r ) ∈ I r , the r labels A i 1 . . . A ir being not necessarily different, and an axiom defined as an initial word of symbols. In [START_REF] Nivoliers | L-system specification of knot-insertion rules for non-uniform B-spline subdivision[END_REF], such an L-system is said to be valid if there exist positive real numbers a i and a ratio ρ strictly larger than 1 such that the rules (2.1) are compatible with the split of intervals with length a (j) i , that is a i = r l=1 (a i l /ρ). If the L-system is valid, then its rules define the subsets K(j) and the subspaces V j := clos span {ϕ j,κ , κ ∈ K(j)} define a multiresolution analysis of S = L ∞ (Ω).

The case of Fibonacci L-system and cubic B-splines

In this article we consider the Fibonacci system which is defined by labels {A 1 = L, A 2 = S} and rules {L → LS, S → L}, with the following compatible lengths {a 1 = 1, a 2 = 1/φ} and ratio ρ = φ, where φ = 1+ √ 5 2 > 1 is the golden number. An infinite word obtained using these rules with {L} as the axiom, is called the Fibonacci word which is known to contain only d + 2 different subwords of length d + 1. This means that, in the case of cubic B-splines scaling functions, at each level j, scaling functions ϕ j,κ are defined as dilates and translates of a set of five different B-splines whose support is made up of intervals labeled as LSLL, SLLS, LLSL, LSLS or SLSL. The eleven other 4-length words, i.e. those containing SS (8) or LLL (3), cannot appear with the proposed rules and thus will never be present provided they are not in the axiom. The grid designed at each level j with such rules are sequences of intervals of length φ -j , labeled by L (for long), and intervals of length φ -j-1 , labeled by S (for short). Let t j,k be the k-th knot at level j, ω ∈ {LSLL, SLLS, LLSL, LSLS, SLSL} be the word defining the length of the support of the scaling function starting at κ = t j,k , and j = 0 be the coarsest level. The scaling functions may be written as:

ϕ j,κ (x) = ϕ ω (φ j (x -t j,k )) = len(ω) 4 M δ ω,0 ,...,δ ω,4 (φ j (x -t j,k )), (2.2)
where δ ω,l := φ j (t j,k+lt j,k ), for l ∈ {0, . . . , 4}, and len(ω

) := δ ω,4 -δ ω,0 = δ ω,4 is the length of the support of ϕ ω : len(LSLL) = len(LLSL) = 3 + φ -1 , len(SLLS) = len(LSLS) = len(SLSL) = 2 + 2φ -1 .
The five mother functions ϕ ω are shown in Figure 1. Note that these scaling functions would have been those designed in [START_REF] Andrle | Bernuau spline wavelets and sturmian sequences[END_REF] if the authors had chosen their so-called minimal case of first category substitution rules instead of those of second category. They call Fibonacci chain a word produced by rules of whichever category. However, the case of Ω being an interval is not considered in [START_REF] Andrle | Bernuau spline wavelets and sturmian sequences[END_REF], and requires to add extra boundary scaling functions. This particular case is handled in §6.

Regular dyadic setting

The same modeling for the well-known regular dyadic case would define an L-system with one label L, one rule L → LL, one length 1, a ratio ρ = 2, and one mother scaling function with a support labeled LLLL and len(LLLL) = 4 shown at the bottom right in Figure 1.

Subdivision scheme

The subdivision scheme associated with these scaling functions was already defined in [START_REF] Nivoliers | L-system specification of knot-insertion rules for non-uniform B-spline subdivision[END_REF]. Nevertheless let us recall the main steps of its definition. Let f ∈ V j be defined as f (x) = κ∈K(j) f j,κ ϕ j,κ (x). Since V j ⊂ V j+1 , there are {f j+1,κ , κ ∈ K(j + 1)} such that f (x) = κ∈K(j+1) f j+1,κ ϕ j+1,κ (x). Stencils of the subdivision scheme associated with the scaling functions write coefficients f j+1,κ as linear combination of coefficients f j,κ .

Let p j,k be the blossom of the polynomial piece of spline function f on [t j,k , t j,k+1 ), that is the d linear function (with values in R) such that p j,k (t, • • • , t) = f (t) on [t j,k , t j,k+1 ). Then, as explained in [Goldman, 1990], we can write with κ = t j,k ,

f j,κ = p j,k (t j,k+1 , • • • , t j,k+d ).
(3.1)

Note that for all m ∈ {k, . . . , k + d}, p j,k (t j,k+1 , • • • , t j,k+d ) = p j,m (t j,k+1 , • • • , t j,k+d ), and if [t j+1,l , t j+1,l+1 ) ⊂ [t j,k , t j,k+1 ), then p j+1,l = p j,k . From (3.1) and the multilinearity of p j,k , we deduce the subdivision stencils.

f j, r f j, q ? | t j,k L | t j,k+1 S | t j,k+2 L | t j,k+3 ? ? | t j+1,l L | t j+1,l+1 S | t j+1,l+2 L | t j+1,l+3 L | t j+1,l+4 S | t j+1,l+5 ? f j+1, 1 f j+1, 2 f j, 6 f j, 5 | t j,k L | t j,k+1 S | t j,k+2 L | t j,k+3 S | t j,k+4 L | t j,k+5 | t j+1,l L | t j+1,l+1 S | t j+1,l+2 L | t j+1,l+3 L | t j+1,l+4 S | t j+1,l+5 L | t j+1,l+6 L | t j+1,l+7 S | t j+1,l+8 f j+1, 3 f j, s f j, 2 f j, 1 | t j,k L | t j,k+1 S | t j,k+2 L | t j,k+3 L | t j,k+4 S | t j,k+5 L | t j,k+6 | t j+1,l L | t j+1,l+1 S | t j+1,l+2 L | t j+1,l+3 L | t j+1,l+4 S | t j+1,l+5 L | t j+1,l+6 S | t j+1,l+7 L | t j+1,l+8 L | t j+1,l+9 S | t j+1,l+10 f j+1, 4 f j+1, 5 f j+1, 6
Figure 2: Configuration around the splitting of LSL, SLS and LL Because neither SS nor LLL can be produced by the Fibonacci rules, only three configurations can occur, corresponding to the splitting of LSL, SLS, and LL (Figure 2). These three configurations give birth to three sets of stencils (3.2) that depend on the kind of intervals making up the support of the associated functions. More precisely, six classes of coefficients can be defined and, for the sake of simplicity, we will sometimes write f j, q with q ∈ { 1 , . . . , 6 } instead of f j,κ with κ ∈ K(j), where f j, 1 is associated with ϕ LSLL , f j, 2 with ϕ SLLS , f j, 3 with ϕ LLSL and the following scaling function is ϕ LSLL , f j, 4 with ϕ LLSL and the following scaling function is ϕ LSLS , f j, 5 with ϕ LSLS , and f j, 6 with ϕ SLSL .

f j+1, 1 = (4 -2φ) f j, q + (2φ -3) f j, r , f j+1, 2 = (5 -3φ) f j, q + (3φ -4) f j, r , f j+1, 3 = 2φ-1 5 f j, 5 + 6-2φ 5 f j, 6 ,        f j+1, 4 = (7 -4φ) f j, 1 + (4φ -6) f j, 2 , f j+1, 5 = 18-11φ 2 f j, 1 + (6φ -9) f j, 2 + 2-φ 2 f j, s , f j+1, 6 = (2 -φ) f j, 2 + (φ -1) f j, s , (3.2)
with ( q , r ) ∈ {( 3 , 1 ), ( 4 , 5 ), ( 6 , 1 )} and s ∈ { 3 , 4 }. Let us detail the equations for one of the two stencils involved in the first configuration:

f j+1, 1 = p j+1,l (t j+1,l+1 , t j+1,l+2 , t j+1,l+3 ) = p j,k (t j+1,l+1 , t j,k+1 , t j,k+2 ) = t j,k+3 -t j+1,l+1 t j,k+3 -t j,k p j,k (t j,k , t j,k+1 , t j,k+2 ) + t j+1,l+1 -t j,k t j,k+3 -t j,k p j,k (t j,k+1 , t j,k+2 , t j,k+3 ) = 1 + φ + φ 2 2φ 2 + φ f j, q + φ 2φ 2 + φ f j, r = (4 -2φ) f j, q + (2φ -3) f j, r .
The last equality comes from the equation satisfied by the golden ratio φ 2 = φ + 1 which allows us to linearize powers of φ. From these stencils and Figure 2, one can write the refinement relations satisfied by the scaling functions:

ϕ LSLL (x) = (2φ -3) ϕ LSLL (φx) + (3φ -4) ϕ SLLS (φx -1) + (7 -4φ) ϕ LLSL (φx -1 -φ -1 ) + 18-11φ 2 ϕ LSLS (φx -2 -φ -1 ), ϕ SLLS (x) = (4φ -6) ϕ LLSL (φx) + (6φ -9) ϕ LSLS (φx -1) + (2 -φ) ϕ SLSL (φx -2), ϕ LLSL (x) = 2-φ 2 ϕ LSLS (φx) + (φ -1) ϕ SLSL (φx -1) + (4 -2φ) ϕ LSLL (φx -1 -φ -1 ) + (5 -3φ) ϕ SLLS (φx -2 -φ -1 ), ϕ LSLS (x) = (2φ -3) ϕ LSLL (φx) + (3φ -4) ϕ SLLS (φx -1) + 2φ-1 5 ϕ LSLS (φx -1 -φ -1 ), ϕ SLSL (x) = 6-2φ 5 ϕ SLSL (φx) + (4 -2φ) ϕ LSLL (φx -φ -1 ) + (5 -3φ) ϕ SLLS (φx -1 -φ -1 ). (3.3)
4 Filter bank and wavelets

From here we do not construct semiorthogonal wavelets as proposed in [Bernuau, 1998, Andrle et al., 2004]. Instead, we focus on the efficiency of the associated multiresolution framework that we want to be a perfect reconstruction filter bank with decomposition and reconstruction defined as simple local linear operators. The subdivision scheme may be implemented as part of a filter bank if the coefficients {f j+1,κ , κ ∈ K(j + 1)} are split into two subsets: new coefficients introduced at level j + 1 called N j and initialized with zeros, and A j initialized with the coefficients at level j (in particular, #A j = #K(j)).

We choose these subsets such that stencils (3.2) define short filters on {f j+1,κ , κ ∈ K(j + 1)}: each coefficient f j,κ is associated with the "central" knot t j,k+2 where κ = t j,k and is changed into f j+1,π where π = t j+1,p , and such that t j,k+2 = t j+1,p+2 . The altered coefficients are f j+1, 1 ← f j, q , f j+1, 2 ← f j, r , and f j+1, 5 ← f j, 2 . The new introduced coefficients are f j+1, 3 , f j+1, 4 and f j+1, 6 . In order to be part of a filter bank implemented as a lifting scheme, the stencils should be decomposed into a succession of elementary steps, implemented in place. An elementary step is defined as follows: {f j+1,κ , κ ∈ K(j + 1)} is partitioned into two subsets W (read and written coefficients) and R (read only coefficients). Each coefficient of W is altered either by a multiplication with a non zero weight, or by an addition with a linear combination of coefficients of R. The elementary steps composing a regular subdivision stencils, are typically defined with subsets (W, R) being alternatively (N j , A j ) and (A j , N j ). In our case, other subsets have to be used in order to deal with the configuration where no new coefficient is inserted between two successive coefficients. Let us present our choice for (W, R) with the factorization of stencils into elementary steps corresponding to the subdivision of a sequence {f j, 5 , f j, 6 , f j, 1 , f j, 2 , f j, s }. This factorization is illustrated in Figure 3 where altered coefficients belonging to A j are drawn with black circles, new coefficients belonging to N j are drawn with white circles and, for each elementary step, coefficient belonging to W are framed in a triangle. Weights are equal to:

   a 35 = 2φ-1 5 , a 36 = 6-2φ 5 ,        b 11 = 4 -2φ, b 22 = φ 2 , c 12 = 10 -6φ, d 21 = 1 -φ 2 ,            a 41 = 7 -4φ, a 42 = 4φ -6, a 62 = 2 -φ = b 54 , a 6s = φ -1 = b 56 , c 55 = 1 2 .
This factorization is not unique and depends in particular on the order of the elementary steps that we apply. Let us detail it with the first two stencils of equation (3.2) where, in the case illustrated by Figure 3, ( q , r ) = ( 6 , 1 ).

     f j+1, 1 = b 11 f j, 6 + c 12 b 22 f j, 1 = (4 -2φ) f j, 6 + (2φ -3) f j, 1 , f j+1, 2 = d 21 b 11 f j+1, 1 + b 22 f j, 1 = d 21 b 11 f j, 6 + (d 21 c 12 b 22 + b 22 ) f j, 1 = (5 -3φ) f j, 6 + (3φ -4) f j, 1 ,        b 11 = (4 -2φ) , d 21 = (5 -3φ) / b 11 , b 22 = (3φ -4) -d 21 (2φ -3) , c 12 = (2φ -3) / b 22 .
From a practical point of view, when the filter bank is implemented in place, a unique vector of coefficients and zeros may be used. In order to apply the elementary steps described above, one should be able to find, in this vector, the first neighbors of a given coefficient within the same resolution j. Let J be the maximum resolution and Fib(j) be the Fibonacci sequence initialized with Fib(0) = Fib(1) = 1. Let ϕ ω be the mother function associated with the coefficient of interest. The left neighbor of this coefficient within the same resolution j, lies in the entry distanced by Fib(Jj + 1) if the second symbol of ω is L, and Fib(Jj) if the symbol is S. The right neighbor is found similarly when considering the third

f j, 5 0 f j, 6 f j, 1 0 f j, 2 0 f j, s a35 1 a36 a41 1 a42 a62 1 a6s b22 b11 b22 b54 1 b56 b11 1 c12 c55 1 f j+1, 2 1 f j+1, 3 f j+1, 1 f j+1, 2 d21 1 f j+1, 4 f j+1, 5 f j+1, 6 f j+1, 1
Figure 3: Subdivision stencils decomposed into in place steps symbol of ω. Note that in the regular dyadic setting, the distance between two coefficients of the same resolution j is 2 J-j . When a subdivision scheme can be decomposed into elementary steps, it is simple to invert it: elementary steps are applied in reversed order, with multiplicative update applied with inverse weights, and additive update applied with weights of opposite sign. From this it is straightforward to build a first multiresolution framework for signal decomposition and reconstruction as presented in [Sweldens, 1998, Schröder and[START_REF] Schröder | [END_REF]. A signal at resolution j is a sequence of numbers λ j := {λ j,κ } κ∈K(j) . The framework is made up of a set of four operators for each resolution. A projection (or decimation) operator Hj : λ j = Hj λ j+1 defines the signal at a coarser resolution. A prediction operator H * j reconstructs the fine signal with errors e j+1 := λ j+1 -H * j λ j . Such an error signal takes non zero values on index set K(j +1) but it can be coded into as a so-called detail signal indexed by M(j) = K(j + 1) \ K(j): γ j := {γ j,m } m∈M(j) . This detail signal is defined from the fine signal with the operator Gj : γ j = Gj λ j+1 . Finally an operator G * j constructs the error signal from the detail signal allowing us to write the perfect reconstruction as λ j+1 = H * j λ j + G * j γ j . These relationships are summarized in Figure 4

H * j G * j Hj Gj = H * j Hj + G * j Gj = I j+1 . (4.1)
This notation is the one used in [Sweldens, 1998] with A * being the adjoint of operator A. It highlights the fact that the prediction operator H * j defined using subdivision stencils, is the adjoint of a projection operator and vice versa.

Operators Hj and Gj constitute the analysis transform (named also decomposition) whereas operators H * j and G * j constitute the synthesis transform (or reconstruction). In order to get the so-called perfect reconstruction property, synthesis followed by analysis must be the identity as well, and the operators must satisfy (4.2) where I j is the identity operator for signals indexed by K(j).

Hj Gj H * j G * j = Hj H * j Hj G * j Gj H * j Gj G * j = I j 0 0 I j . (4.2)
When equations (4.1) and (4.2) are satisfied, the operators are said to be biorthogonal. The subdivision scheme designed in §3 defines a prediction operator H * j . The filter bank defined by its implementation into elementary steps as described in Figure 3 and its inverse, both applied to signals λ j and λ j+1 instead of coefficients {f j,κ , κ ∈ K(j)} and {f j+1,κ , κ ∈ K(j + 1)}, and with the coefficients of a detail signal γ j replacing the zeros of N j , define such four biorthogonal operators.

For later purpose, we now translate the notation of operators H * j , G * j into the following linear equation:

λ j+1,ℓ = κ∈K(j) h j,κ,ℓ λ j,κ + m∈M(j) g j,m,ℓ γ j,m , (4.3)
where h j,κ,ℓ are also involved into the refinement relations (3.3) satisfied by the non-uniform cubic B-spline scaling functions ϕ j,κ = ℓ∈K(j+1) h j,κ,ℓ ϕ j+1,ℓ . Equation ( 4.3) defines also a basis of the complementary space W j between two successive subspaces V j ⊕ W j = V j+1 with wavelet functions {ψ j,m , m ∈ M(j)} defined as

ψ j,m = ℓ∈K(j+1) g j,m,ℓ ϕ j+1,ℓ . (4.4)
Like similar constructions, these first wavelets have poor properties and the whole framework can be improved by lifting [Sweldens, 1998, Schröder and[START_REF] Schröder | [END_REF].

Update by lifting

Lifting means, as illustrated by Figure 5, adding another elementary step at the beginning of the synthesis filter bank decomposition described in Figure 3, where each detail coefficient γ j,m (remember these occupy the zeros location on that figure) diffuses on samples in its vicinity. In order to keep the whole framework biorthogonal, the inverse of this update elementary step must be added at the end of the analysis filter bank. By doing so, one changes projection operator Hj and operator G * j , but not prediction operator H * j or operator Gj .

A biorthogonal framework with good properties is efficient, meaning made of short filters implemented in place, and stable, that is, allowing robust reconstruction and decomposition with respect to small value changes due to rounded computation (the golden ratio is an irrational number) or lossy transmission. Note that by construction the lifting structure of the filter bank ensures that the implementation can be done in place. The question of stability has been addressed carefully by Dahmen et al., in particular in [Dahmen, 1994, Carnicer et al., 1996]. Ideally the condition numbers of transformation T relating the fine scale data to their multiscale representation and its inverse transform should remain uniformly bounded. This is equivalent to ask that the union of the basis functions of V 0 and the wavelets at all levels constitute a Riesz basis of S; this is called stability over all levels. Necessary and sufficient conditions to get such a property are threefold: the analysis and synthesis transforms are uniformly bounded, they constitute a biorthogonal set of filters at each level, and the primal and dual subdivision schemes converge. If the set of filters are biorthogonal and the primal subdivision scheme converges by construction, to show that the dual non-uniform subdivision scheme converges is a difficult task.

To get stability over all levels another solution would be to define full orthogonal wavelet functions: the transformation T is then orthogonal. However, the construction of such orthogonal multiscale bases can lead to problematic inversion and are often associated with very large filters. To impose some kind of stability in a non orthogonal framework, a natural approximation is to keep the orthogonality between successive levels (W j ⊥V j ) but to relax the orthogonality between wavelets at the same level; this is called semiorthogonality [Quak, 2002, Bernuau, 1998, Andrle et al., 2004]. As explained in the introduction, this is not satisfactory for our aim to keep a local approach and to define the wavelets with a finite number of different filters. To achieve this goal, a first solution consists in approximating semiorthogonality by defining wavelets basis functions in W j that are orthogonal to only some functions in V j . As sumed up in [START_REF] Maes | Stability analysis of biorthogonal multiwavelets whose duals are not in and its application to local semiorthogonal lifting[END_REF], at this stage, two strategies to define the update are possible and these can be mixed: local semiorthogonal lifting [Lounsbery et al., 1997, Simoens and[START_REF] Simoens | [END_REF] and wavelets with vanishing moments [Sweldens, 1998]. As already noted in [START_REF] Kobbelt | A multiresolution framework for variational subdivision[END_REF]], each of these strategies leads to wavelets orthogonal to a different set of functions in V j , being respectively a subset of the basis scaling functions ϕ j,k , and some low degree polynomials. However, all sets of functions are not of equal importance for stability as first vanishing moment is considered as mandatory. Another solution consists in removing only one knot at each step as proposed by Bertram and then enforce a discrete orthogonalization instead of orthogonalization of continuous functions [Bertram, 2005]. We prefer to keep the possibility of removing several knots within each step, and choose consequently the solution based on the vanishing moments of wavelets. More precisely, we ask the wavelets to be orthogonal to x → x p , 0 ≤ p < Ñ . Combination with local semiorthogonality is left for future work. Note that if dual wavelets existed, then they would be orthogonal to {ϕ j,κ }, and so to x → x p , 0 ≤ p < N = d + 1 = 4.

Regular dyadic setting In the regular dyadic case, this modeling would yield to the Cohen-Daubechies-Feauveau (N, Ñ ) biorthogonal wavelets [START_REF] Cohen | Biorthogonal bases of compactly supported wavelets[END_REF].

Wavelets as combination of fine scaling functions

Equation (4.4) enables us to write wavelets at level j as a combination of known B-spline scaling functions at level j + 1. As explained in [START_REF] Sweldens | Building your own wavelets at home[END_REF], the weights involved in such a refinement relation for a given ψ j,m , m ∈ M(j), can be obtained with our filter bank including the update step (Figure 5) by feeding the synthesis part of the filter bank with null values in every location except γ j,m = 1. Then, the resulting signal λ j+1 contains the weights {g j,m,l , l ∈ K(j + 1)}. In order to satisfy the Ñ equations corresponding to the expected Ñ vanishing moments of wavelet functions, we use Ñ -wide update filters. It follows that, as in the regular case [START_REF] Cohen | Biorthogonal bases of compactly supported wavelets[END_REF], the support of each wavelet ψ j,m is made up with N + Ñ -1 intervals at level j. This support is associated with a particular word ω whose central symbol is always L, label of the interval whose splitting creates the detail coefficient γ j,m . The labels of the support of the scaling functions involved in (4.4) are the subwords of length 4 of ω after application of the L-system rules. Moreover, if the support begins at t j,k then the wavelet can be written as ψ j,m (x) = ψ ω (φ j (xt j,k )).

(5.1)

In the same line as what we did with classes of coefficients f j,κ , we define classes of detail coefficients, but with the word ω as subscript: γ j,ω .

In this section, we detail the case Ñ = 2. Only three words of N + Ñ -1 = 5 symbols with a central L are allowed in the word of Fibonacci, as containing neither SS nor LLL: LS -L -SL, LS -L -LS and SL-L-SL (we emphasize the central L). Therefore we define three classes of detail coefficients and one 2-wide update filter per class: [α -1 , α +1 ] associated with γ j,LS-L-SL , [β -1 , β +1 ] with γ j,LS-L-LS , and [ν -1 , ν +1 ] with γ j,SL-L-SL . Using definition of mother scaling (2.2) and wavelet functions (5.1), as well as Figure 5, we write ψ j,LS-L-SL as a combination of mother scaling functions associated with the five update subdivision scheme

λ j, 4 λ j, 5 γj,LS-L-SL λ j, 6 λ j, 1 1 1 α-1 α+1 1 1 a35 1 a36 b11 b22 b11 b22 1 c12 1 c12 λ j+1, 1 λ j+1, 2 d21 1 λ j+1, 3 λ j+1, 1 λ j+1, 2 d21 1 update subdivision scheme λ j, 3 or λ j, 6 λ j, 1 γj,LS-L-LS λ j, 2 γj,SL-L-SL λ j, 3 or λ j, 4 λ j, 1 or λ j, 5 1 1 β-1 β+1 1 ν-1 ν+1 1 1 a41 1 a42 a62 1 a6s b11 b22 b54 1 b56 b11 b22 1 c12 c55 1 c12 λ j+1, 1 λ j+1, 2 d21
ψ LS-L-SL (x) = [c 12 b 22 α -1 ] ϕ LSLL (φx) + [(1 + d 21 c 12 ) b 22 α -1 ] ϕ SLLS (φx -1) + [1 + a 35 α -1 + a 36 α +1 ] ϕ LLSL (φx -1 -φ -1 ) + [b 11 α +1 ] ϕ LSLL (φx -2 -φ -1 ) + [d 21 b 11 α +1 ] ϕ SLLS (φx -3 -φ -1 ) = [(2φ -3) α -1 ] ϕ LSLL (φx) + [(3φ -4) α -1 ] ϕ SLLS (φx -1) + 1 + 2φ-1 5 α -1 + 6-2φ 5 α +1 ϕ LLSL (φx -1 -φ -1 ) + [(4 -2φ) α +1 ] ϕ LSLL (φx -2 -φ -1 ) + [(5 -3φ) α +1 ] ϕ SLLS (φx -3 -φ -1 ).
(5.2)

Similarly, we write for the two other classes:

ψ LS-L-LS (x) = [(2φ -3) β -1 ] ϕ LSLL (φx) + [(3φ -4) β -1 ] ϕ SLLS (φx -1) + [1 + (7 -4φ) β -1 + (4φ -6) β +1 ] ϕ LLSL (φx -1 -φ -1 ) + 2-φ 2 + 18-11φ 2 β -1 + (6φ -9)β +1 ϕ LSLS (φx -2 -φ -1 ) + [(2 -φ) β +1 ] ϕ SLSL (φx -3 -φ -1 ),
(5.3)

ψ SL-L-SL (x) = [(4φ -6) ν -1 ] ϕ LLSL (φx) + φ-1 2 + (6φ -9) ν -1 + 2-φ 2 ν +1 ϕ LSLS (φx -1) + [1 + (2 -φ) ν -1 + (φ -1) ν +1 ] ϕ SLSL (φx -2) + [(4 -2φ) ν +1 ] ϕ LSLL (φx -2 -φ -1 ) + [(5 -3φ) ν +1 ] ϕ SLLS (φx -3 -φ -1 ).
(5.4)

Vanishing first moments of wavelets

As stated in [Neuman, 1981] for distinct knots and generalized in [de Boor, 1976] for t j,k < t j,k+d+1 when B-splines of order d + 1 are considered, the moments of the B-spline of Curry and Schoenberg M t j,k ,...,t j,k+d+1 defined on the intervals between t j,k and t j,k+d+1 with p ≥ 0 are:

x p M t j,k ,...,t j,k+d+1 (x) dx = (d + 1)! p! (d + 1 + p)! k≤k 1 ≤k 2 ≤...≤kp≤k+d+1 t j,k 1 t j,k 2 . . . t j,kp .
In particular, the B-spline function ϕ ω defined by (2.2) satisfy

x p ϕ ω (x) dx = len(ω) 4 4! p! (4 + p)! µ ω,p with µ ω,p := 0≤k 1 ≤k 2 ≤...≤kp≤4 δ ω,k 1 δ ω,k 2 . . . δ ω,kp .
As a consequence, with (l, s) ∈ Z,

x p ϕ ω (φx -l -sφ -1 ) dx = len(ω) 4φ p+1 p r=0 p r (l + sφ -1 ) p-r 4! r! (4 + r)! µ ω,r .
(5.5)

In particular for p = 0, µ ω,0 = 1 whatever ω, and ϕ ω (φxlsφ -1 ) dx = len(ω) 4φ which leads to, with equations (5.2), (5.3): and (5.4):

4φ ψ LS-L-SL (x)dx = (2φ + 2) α -1 + (φ + 2) + (2φ + 2) α +1 , 4φ ψ LS-L-LS (x)dx = (3φ + 1) β -1 + (2φ + 1) + (2φ + 2) β +1 , 4φ ψ SL-L-SL (x)dx = (2φ + 2) ν -1 + (2φ + 1) + (3φ + 1) ν +1 .
From (5.1), first moment of ψ j,m vanishes if first moment of the associated function ψ ω does and conditions for vanishing first moments of wavelets are:

   (φ -3) = 2 α -1 + 2 α +1 , φ = (1 -2φ) β -1 -2 β +1 , φ = -2 ν -1 + (1 -2φ) ν +1 .
(5.6)

Vanishing second moments of wavelets

From (5.5), for any (l, s) ∈ Z 2 :

xϕ ω (φx -l -sφ -1 ) dx = len(ω) 4φ 2 (l + sφ -1 + 1 5 µ ω,1 ),
where µ SLSL,1 = 6φ -2, µ LSLL,1 = 3φ + 4, µ SLLS,1 = 5φ, µ LLSL,1 = 2φ + 6, and µ LSLS,1 = 4φ + 2. As a consequence,

20φ 2 x ψ LS-L-SL (x) dx = (32φ + 20) α -1 + (27φ + 19) + (48φ + 30) α +1 , 20φ 2 x ψ LS-L-LS (x) dx = (37φ + 24) β -1 + (34φ + 21) + (50φ + 30) β +1 , 20φ 2 x ψ SL-L-SL (x) dx = (30φ + 20) ν -1 + (32φ + 19) + (53φ + 31) ν +1 .
From (5.1), if first and second moments of ψ ω vanish, then second moment of ψ j,m vanishes as well. With (5.6), we get the values of the update filters which make first and second moments of wavelets ψ j,m vanish:

α -1 = 11 -7φ, α +1 = 5 2 (3φ -5), β -1 = 5 (4 -3φ), β +1 = 1 2 (4 -3φ), ν -1 = φ -2, ν +1 = φ -2.
The three mother functions ψ ω are shown in Figure 6.

More vanishing moments of wavelets

With the same procedure, we can compute with our filter bank equations similar to (5.2), (5.3) and (5.4) for any width Ñ of update filters. With (5.5) we can deduce equations similar to (5.6) for each power p ∈ {0, . . . , Ñ -1}, making up a linear system which can be solved numerically with a linear algebra library such as GSL. Update filters and resulting wavelet functions are available as supplementary materials linked from the main article webpage. Note that the number of different filters needed does not grow quickly with Ñ : three filters are needed for Ñ = 2, five filters for Ñ = 4 and seven filters for Ñ ∈ {6, 8}.

Boundaries

In [Andrle, 2002, Andrle et al., 2004], the authors deal with infinite domain centered on the origin. With our Fibonacci L-system, they use these substitution rules, and so original functions, on the right side of the domain, while mirrored rules and functions are used on the left. A finite number of functions whose definition domain overlaps the two sides, remains possibly to be designed according to the chosen degree.

The two extremities of the domain are not considered. Such a modeling is used to analyze quasicrystal diffraction patterns.

On the contrary, we want to design a framework which can be applied in practice and efficiently to signals defined on the interval, and the question of domain boundaries has to be handled. Different solutions can be investigated, either starting with adapting the framework filters as proposed in [START_REF] Sweldens | Building your own wavelets at home[END_REF], or defining new multiresolution spaces of functions living in the interval. We propose a solution which belongs to the second category. As what has been done by many authors to adapt uniform B-spline semiorthogonal [START_REF] Chui | Wavelets on a bounded interval[END_REF] or biorthogonal [Bittner, 2006, Primbs, 2010, Černá and Finěk, 2011] wavelets to an interval, B-splines with multiple knots at endpoints of the interval may be introduced as a complement to inner ordinary B-spline scaling functions. Quak used also multiple knot B-splines as boundary scaling functions in his construction of semiorthogonal non-uniform B-spline wavelets [Quak, 2002]. Note that other adaptations of uniform B-spline wavelets to the interval preferred to define boundary scaling functions as combinations of truncated simple knot B-splines [START_REF] Cohen | Wavelets on the interval and fast wavelet transforms[END_REF], Dahmen et al., 1999].

L-system and scaling functions

We extend the framework described in the previous sections by adding to the Fibonacci L-system a new label X associated with the rule X → X and a length equal to zero. Besides, the axiom ω 0 becomes X d ω 0 X d , with B-splines of order d + 1. In our case, d = 3 and we choose ω 0 = L. If ω j is the word of symbols of the intervals after j subdivision steps with the previous framework, ω j is a subword of Fibonacci, meaning that ω j+1 = ω j-1 ω j . The word of symbols in the new framework after j steps is X d ω j X d and the number of samples in resolution j is equal to 3 + Fib(j + 1). Note that in the regular dyadic setting, this number of samples is equal to 3 + 2 j .

Regarding scaling functions, since the first word is X 3 LX 3 , four scaling functions are used in the first resolution j = 0: ϕ XXXL , ϕ XXLX , ϕ XLXX and ϕ LXXX shown in Figure 7 (top-left function, rightest function in the second row and the two first functions in the last row). The second word is X 3 LSX 3 , defining five mother scaling functions for resolution j = 1: ϕ XXXL , ϕ XXLS , ϕ XLSX , ϕ LSXX , and ϕ SXXX shown in Figure 7 (the two first functions in the first row, the two last functions in the second row and the last function in the bottom row).

For j > 1, the word ω j starts always as X 3 LSL . . ., yielding to the three functions in the first row in Figure 7 as the first mother scaling functions used near the left boundary. However, the word ω j finishes alternatively as . . . LSLX 3 or . . . LLSX 3 : in even resolution j, the mother scaling functions shown in the second row in Figure 7 will be used near the right boundary, whereas for odd j the functions in the third row will be chosen. In total, twelve new mother scaling functions are introduced for managing boundaries.

Let us illustrate how the blossom-based construction of subdivision stencils presented in §3, adapts to such 0-length knot intervals. With the same notation, we can write (Figure 8): f j+1,t j+1,0 = p j+1,0 (t j+1,1 , t j+1,2 , t j+1,3 ) = p j,0 (t j,1 , t j,2 , t j,3 ) = f j,t j,0 , f j+1,t j+1,1 = p j+1,1 (t j+1,2 , t j+1,3 , t j+1,4 ) = p j,1 (t j,2 , t j,3 , t j+1,4 ) = t j,4t j+1,4 t j,4t j,1 p j,1 (t j,1 , t j,2 , t j,3 ) + t j+1,4t j,1 t j,4t j,1 p j,1 (t j,2 , t j,3 , t j,4 ) The twelve mother scaling functions ϕ ω with plain red parts defined above L intervals and blue dashed parts defined above S intervals, used near the boundaries with Ñ = 2.

= 1 φ 2 f j,t j,0 + φ φ 2 f j,t j,1 = (2 -φ) f j,t j,0 + (φ -1) f j,t j,1 .

Update filters and wavelet functions

We follow the same process as the one described in previous sections, leading to filter banks implemented in place and wavelets with vanishing moments. Figures 8,9, and 11 illustrate the synthesis filter banks for the different boundary neighborhoods considered above, and with an update elementary step designed in order to get wavelet functions with two vanishing moments. Note that for simplicity, we name in these figures the introduced classes of coefficients and details as λ j,ω instead of λ j, q . The associated wavelet functions satisfy the following equations, and are illustrated in Figure 10:

ψ XX-L-SL (x) = (6 -4φ) ϕ XXLS (φx) + (4φ -6) ϕ XLSL (φx) + (16φ -26) ϕ LSLL (φx) + (21φ -34) ϕ SLLS (φx -1) ψ LS-L-XX (x) = (21φ -34) ϕ LSLL (φx) + (29φ -47) ϕ SLLS (φx -1) + (7φ -11) ϕ LLSX (φx -1 -φ -1 ) + (11 -7φ) ϕ LSXX (φx -2 -φ -1 ) ψ SL-L-SX (x) = 3-2φ 2 ϕ LLSL (φx) + 10-5φ 4 ϕ SLSL (φx -2) + 4-3φ 2 ϕ LSLX (φx -2 -φ -1 ) ψ XX-L-XX (x) = (3 -2φ) ϕ XXLS (φx) + (4φ -6) ϕ XLSX (φx) + (3 -2φ) ϕ LSXX (φx) ψ XX-L-SX (x) = (φ -2) ϕ XXLS (φx) + (4φ -6) ϕ XLSL (φx) + (3 -2φ) ϕ LSLX (φx)
On the contrary with wavelets defined far from the boundaries, these wavelet functions are not defined as the sum of five scaling functions. Indeed, due to the presence of X in the label of the support of the wavelet function, the word produced by the L-system rules contains less than five 4-length factors. For example, XXLSL → XXLSLLS contains only four 4-length factors.

update subdivision scheme

λ j,X 3 L λ j,X 2 LS γj,XX-L-SL λj,XLSL λ j, 1 1 2φ -4 3φ -5 1 1 φ -1 1 2 -φ φ -1 b11 b22 1 c12 λ j+1,X 3 L λ j+1,X 2 LS 2 -φ 1 λj+1,XLSL λ j+1, 1 λ j+1, 2 d21 1
Figure 8: Lifting scheme for synthesis near the left boundary for j > 1

λ j, 6 λj,LSLX γj,LS-L-XX λ j,SLX 2 λ j,LX 3 1 1 8 -5φ 4 -3φ 1 2φ -3 1 4 -2φ b11 b22 2 -φ 1 c12 1 φ -1 λ j+1, 1 λ j+1, 2 d21 1 λj+1,LLSX λ j+1,LSX 2 λ j+1,SX 3 update subdivision scheme γ j, 2 λ j, 2 γj,SL-L-SX λj,LLSX λ j,LSX 2 λ j,SX 3 β+1 1 -1 4 φ-3 2 1 1 a42 a62 1 a6s b54 1 b56 φ -1 c55 1 2 -φ λ j+1, 4 λ j+1, 5 λ j+1, 6
λj,LSLX λ j,SLX 2 λ j,LX 3 The same procedure can be applied to design update filters defining wavelet functions with Ñ > 2 vanishing moments. However, as the filter width grows with Ñ , the level must be greater than the minimum level which provides enough neighbors (for instance j >= 2 if Ñ = 6), and precautions have to be taken near the boundaries. If Ñ ≥ 6, some filters may not expand equally on the left and on the right of the central L interval. We choose to keep the filter width as short as possible and use the first following right neighbors to replace the neighbors which do not exist on the left, and vice versa near the right boundary of Ω. This choice implies that several wavelet functions, such as ψ XXX-L-SLLSLSL and ψ XXXLS-L-LSLSL , may be defined as a linear combination of the same scaling functions. But due to the different position of the central L within the common ω, these functions are well distinct. Note that, due to the presence of S intervals, these cases are less numerous in the Fibonacci setting than in the regular dyadic setting: they occur only for Ñ ≥ 8 with only two wavelet functions per boundary defined with the same scaling functions within each level. In the regular dyadic setting, this occurs from Ñ = 6, with Ñ /2 -1 wavelet functions per boundary (in general, ( Ñ -N )/2 + 1). Update filters for Ñ ∈ {4, 6, 8} are

λ 0,X 3 L λ 0,X 2 LX γ0,XX-L-XX λ 0,XLX 2 λ 0,LX 3 1 φ -2 1 1 -φ 2 -φ 1 φ -1 φ -1 2 -φ λ 1,X 3 L λ 1,X 2 LS 2 -φ 1 λ1,XLSX λ 1,LSX 2 1 φ -1 λ 1,SX 3 λ 1,X 3 L λ 1,X 2 LS γ1,XX-L-SX λ1,XLSX λ 1,LSX 2 λ 1,SX 3 1 1 -φ 1 φ -2 φ -1 1 2 -φ φ -1 φ -1 λ 2,X 3 L λ 2,X 2 LS 2 -φ 1 λ2,XLSL λ2,LSLX 1 2 -φ λ 2,SLX 2 λ 2,LX 3
Figure 11: Lifting scheme for synthesis from j = 0 to 1 (to the left) or from j = 1 to 2 (to the right) available as supplementary materials linked from the main article webpage.

Regular dyadic setting In the regular dyadic case, our construction to deal with boundaries would yield to the same wavelet functions as Type A wavelets built by Kai Bittner [Bittner, 2006].

Stability

In §5, we recalled the necessity for our multiresolution framework to be as stable as possible, with the desirable property of stability over all levels being subordinate to the convergence of the dual subdivision scheme, which is difficult to prove since the scheme in non-stationary. In this section, we give an insight into this stability through numerical analysis of condition number of the framework. But first of all, we remark that our framework fulfills the conditions for a weaker definition of stability proposed in [START_REF] Carnicer | Local decomposition of refinable spaces and wavelets[END_REF]: the uniform stability of {Φ j ∪ Ψ j }, with Φ j := { φj,κ , κ ∈ K(j)} and Ψ j := {ψ j,m , m ∈ M(j)}.

Uniform stability

Let A j be the synthesis operator from K(j + 1) onto K(j + 1) which transforms λ j ∪ γ j into λ j+1 and B j the analysis operator which, by construction, is the inverse of A j . The sets Φ j and Ψ j satisfy the hypotheses of Corollary 2.1 in [START_REF] Carnicer | Local decomposition of refinable spaces and wavelets[END_REF], and then {Φ j ∪ Ψ j } is said uniformly stable if and only if

A j = O(1), B j = O(1), j → ∞.
Whatever the update filter chosen in order to get wavelets with vanishing moments, there is only a finite number of different rules to be applied. Let us consider the norm of the operators defined by the infinity norm of their matrix representation which will be noted also as A j and B j for simplicity. The weights included in each line of A j or B j depend on a factor of a given length l in X d ω j X d . As we know that the number of different factors of length l in the infinite word of Fibonacci is l + 1 and because l does not depend on j, the number of different possible absolute row sums in A j or B j is finite, which leads to the result.

To be more specific, numerical experiments give A j ∞ ≈ 2 and B j ∞ ≈ 4 whatever Ñ , yielding to a condition number A j ∞ • B j ∞ < 10 for any resolution j which is large enough (greater than 1 to 6, depending on Ñ ∈ {0, 2, 4, 6, 8}). Indeed, ω j+1 = ω j-1 ω j means that once ω j 0 contains all the possible factors (without X) of a given length, the following ω j do so. Moreover, the l-length factors starting with an X are the same whatever j as soon as 3 + Fib(j + 1) ≥ l, when the l-length factors ending with X are the same every two resolutions.

Note that the boundary management does not alter too much this one-step stability. Indeed, if we compute the maximum absolute value row sum of B j and A j without the rows which contains a weight defined for the boundary management, the norms remain of the same order. Besides, these values are similar to the norms computed with similar frameworks designed with the regular dyadic hierarchy instead of the Fibonacci tiling.

Stability over all levels

The stability over all levels defined in [START_REF] Carnicer | Local decomposition of refinable spaces and wavelets[END_REF] is closely related to the condition of the framework defined as follows. Let T j be the synthesis operator from K(j + 1) onto K(j + 1) which transforms λ 0 ∪ j k=0 γ k into λ j+1 . The operators T j are well conditioned if

T j = O(1), T -1 j = O(1), j → ∞.
As for A j and B j , the weights in any row of T j or T -1 j are determined by a factor of a given length l j in X d ω j X d . Unfortunately, in contrast with l, the length l j grows with j, which prevents us to conclude to the stability over all levels. However, numerical experiments show that this instability is of the same order as the instability of the framework designed with the regular dyadic setting.

Let T j 0 ,∆ be the synthesis matrix which transforms λ j 0 ∪ j 0 +∆ k=j 0 γ k into λ j 0 +∆ . T -1 j 0 ,∆ is the analysis matrix. Figure 12 (top left) shows the values of the condition number against ∆, for different update lifting corresponding to Ñ ∈ {0, 2, 4, 6, 8}, and with j 0 = 3 which is the first level where the update filters for Ñ = 8 can be applied. As expected, the framework is more stable as Ñ grows. However the condition numbers for Ñ = 8 are slightly higher than for Ñ = 6. This counter-intuitive order appears with the rules added for managing the boundaries. Indeed, let us note T j 0 ,∆ and T -1 j 0 ,∆ the matrices without the rows containing at least one entry which depends on the boundary management. Note that the one is not the inverse of the other. However, with the same argument as the one used for the uniform stability, when j 0 becomes large enough for these submatrices to contain all the possible set of weights within a row, their infinity norms are also the norms of the bi-infinite matrices corresponding to the framework applied on signals without boundary. Figure 12 (top right) shows the conditions numbers of these matrices with j 0 = 6 which yields to norms numerically stable against j ≥ j 0 , whatever Ñ and ∆. The same general behavior of the condition numbers can be observed with, this time, the curve corresponding to Ñ = 8 strictly below the curve for Ñ = 6 for ∆ ≥ 5. The bottom row of the same figure shows the same quantities computed with the framework designed with the regular dyadic setting, and values for j 0 and ∆ chosen in order to get similar numbers of samples at the finest resolution to those got in the experiments illustrated in the top row. The behavior and values of the quantities are very similar between the two rows. This can be explained by the low discrepancy between interval lengths of our non-uniform tiling, outside the boundaries neighborhood: within each resolution, the ratio between interval lengths is never greater than the golden ratio (1.618...).

Conclusion

In this paper, we have presented a new multiresolution analysis based on non uniform B-splines and defined on the Fibonacci grid. Similar constructions are possible with other valid L-system than Fibonacci substitution rules as far as the associated subdivision scheme can be decomposed into elementary steps. Define conditions on the L-system for such a decomposition to be possible is left for future work. Since we focus on the study of a signal supported on an interval, after having explained how to perform its decomposition at inner locations, the emphasis is put on the definition of boundary wavelets and scaling functions. In our framework, the analysis/synthesis steps are naturally associated with biorthogonal filters which allow the decomposition to be computed in place. Furthermore, since the downsampling rate between each level of resolution is lower than 2 and the filter bank is designed on non-uniform samples, we believe that the proposed decomposition could have good properties for further applications in particular in regard to aliasing. This aspect will be the scope of future work. Moreover, the analysis could be extended to image decomposition using a tensor-product approach and the properties of such an encoding should be investigated. Finally, we carried a numerical study showing that the proposed multiscale representation is as stable as the same analysis implemented on a dyadic grid. Alternative strategies combining vanishing moments and local semiorthogonalization could improve the condition number of the framework [START_REF] Simoens | A stabilized lifting construction of wavelets on irregular meshes on the interval[END_REF]. Furthermore, the stability of the decomposition is not theoretically guaranteed yet, because the existence of the dual bases remains to be proven. However the lifting structure of the framework makes the first step of this study, that is, the definition of the dual non-stationary subdivision scheme, easier. But the proof of its convergence is a much more difficult task.

Figure 1 :

 1 Figure1: The five mother scaling functions ϕ ω with plain red parts defined above L intervals and blue dashed parts defined above S intervals. At the bottom right, the mother scaling function ϕ LLLL used in the regular setting

Figure 4 :

 4 Figure 4: Filter bank for multiresolution analysis and synthesis

Figure 5 :

 5 Figure 5: Lifting scheme for synthesis

Figure 6 :

 6 Figure 6: The three mother wavelet functions ψ ω for Ñ = 2 with plain red parts defined above L intervals and blue dashed parts defined above S intervals. At the bottom right, the mother wavelet function ψ LL-L-LL used in the regular setting

  Figure7: The twelve mother scaling functions ϕ ω with plain red parts defined above L intervals and blue dashed parts defined above S intervals, used near the boundaries with Ñ = 2.

Figure 9 :

 9 Figure 9: Lifting scheme for synthesis near the right boundary for j > 1 and even (to the left) or odd (to the right)

Figure 10 :

 10 Figure10: The five mother wavelet functions ψ ω used near boundaries, for Ñ = 2, with plain red parts defined above L intervals and blue dashed parts defined above S intervals.

Figure 12 :

 12 Figure12: Condition numbers of ∆ successive steps of our Fibonacci framework (top row) and of the similar framework with the regular dyadic setting (bottom row), for Ñ ∈ {0, 2, 4, 6, 8}. On the left, the norms are computed with all the rows. On the right, rows depending on the boundary management are excluded.