Multiscale Autoregressive Models and Wavelets Khalid Daoudi, Austin B. Frakt, Student Member, IEEE, and Alan S. Willsky, Fellow, IEEE Abstract-The multiscale autoregressive (MAR) framework was introduced to support the development of optimal multiscale statistical signal processing. Its power resides in the fast and flexible algorithms to which it leads. While the MAR framework was originally motivated by wavelets, the link between these two worlds has been previously established only in the simple case of the Haar wavelet. The first contribution of this paper is to provide a unification of the MAR framework and all compactly supported wavelets as well as a new view of the multiscale stochastic realization problem. The second contribution of this paper is to develop wavelet-based approximate internal MAR models for stochastic processes. This will be done by incorporating a powerful synthesis algorithm for the detail coefficients which complements the usual wavelet reconstruction algorithm for the scaling coefficients. Taking advantage of the statistical machinery provided by the MAR framework, we will illustrate the application of our models to sample-path generation and estimation from noisy, irregular, and sparse measurements.

Index Terms-Fractional Brownian motion, graphical models, internal models, multiscale estimation, multiscale models, stochastic realization, wavelets.

I. INTRODUCTION

I N THE last two decades, multiresolution decomposition techniques, such as the wavelet transform, have been widely and successfully applied in signal processing. This is due both to their ability to compactly capture the salient scale-to-scale properties that many signals exhibit and to the efficiency of the algorithms to which they lead. With both of these attractive features in mind, the multiscale autoregressive (MAR) framework was introduced [START_REF] Basseville | Modeling and estimation of multiresolution stochastic processes[END_REF] to support the development of optimal multiscale statistical signal processing. Like the wavelet transform, MAR processes are recursive in scale and, due to the nature of the scale recursion, fast statistical signal processing algorithms (sample-path generation, linear least squares estimation, likelihood calculation) for MAR processes exist. The power of the MAR framework resides in its ability to simultaneously address several complications which arise in a variety of signal processing applications. For instance, the data sets can be large, the processes to be estimated can be nonstationary, the measurements may be irregularly spaced, nonlocal, and corrupted by nonstationary noise.

Despite the apparent similarities between wavelets and the MAR framework, it seemed that the two could not be easily reconciled except in the simplest case of the Haar wavelet.

The crux of the problem, which this paper resolves, hinges on the particular properties that useful MAR processes possess.

One such property, which is very similar to wavelet reconstruction, is that the finest-scale signal of an MAR process is formed from a coarse signal by successively adding detail. Specifically, the MAR framework provides an implicit secondorder statistical model for a fine-scale signal by creating a dynamically coupled hierarchy of vector-valued signals above the fine-scale signal (see Fig. 1).

The coarse-to-fine dynamics of an MAR process 1, for some integer , are analogous to those of a state-space model [START_REF] Abry | The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and fast implementation[END_REF] where indicates the integer part and is white noise with auto-covariance (see Fig. 1 for a concise summary of our indexing notation). Note, however, that (1) is more general than a classical state-space model because here, is indexed by the nodes of any tree rather than by the integers which form a monadic tree. We call the vector an MAR state.

The form of (1) suggests that MAR dynamics can be made to mimic those of the wavelet reconstruction algorithm. This has been previously shown only in the case of the Haar wavelet [START_REF] Fieguth | Fractal estimation using models on multiscale trees[END_REF]. We will show that through a particular definition of the state vector , the MAR dynamics can be chosen to match the reconstruction algorithm associated with any compactly supported orthogonal or biorthogonal wavelet. We emphasize, however, that signal synthesis is not our purpose. Instead, modeling is our objective. Specifically, given the statistics of a random signal, which we view as indexed by the leaf nodes of a tree, we focus on building an MAR model to capture those given statistics with high fidelity.

A particular class of MAR processes we are going to focus on are internal MAR processes. While we will elaborate on the notion of internality later, briefly an internal MAR process is one for which the state at every node 2is a linear functional of the states which reside at the fine-scale nodes which descend from . In general, there are no constraints on how the MAR states should be defined, but internality is a property which is very useful for many reasons. First, internal models are intellectually important in the context of statistical modeling and are widely used in stochastic realization theory for time-series [START_REF] Lindquist | On the stochastic realization problem[END_REF]. It is, therefore, natural to consider the extension of this idea to the context of MAR processes as is done in [START_REF] Frakt | Computationally Efficient Stochastic Realization for Internal Multiscale Autoregressive Models[END_REF]- [START_REF] Irving | A canonical correlations approach to multiscale stochastic realization[END_REF]. Fig. 1. An MAR process on a dyadic tree. The root node state is x 0 (0). The parent of state x j (n) is x j01 ([n=2]); while its children are x j+1 (2n) and x j+1 (2n + 1).

Second, in many applications there is a need to fuse measurements taken at different scales [START_REF]A multiresolution methodology for signal-level fusion and data assimilation with applications to remote sensing[END_REF]. Frequently, the nonlocal coarse-scale measurements in a multiscale measurement set are linear functionals of the finest scale variables (e.g., tomographic measurements). Since internal MAR models include as coarse states nonlocal functions of the finest scale, they allow efficient fusion of nonlocal and local measurements with no increase in complexity as compared to the case of fusing only fine-scale data.

Finally, as we will see, internality provides a convenient parameterization of the information content of the MAR states. Using this parameterization leads to MAR dynamics that incorporate a powerful optimal prediction of a child state from its parent. This optimal prediction will have important and significant consequences for our ability to accurately model signals using MAR processes based on wavelets.

In the early attempts to marry MAR processes and wavelets, it was incorrectly thought that the internal property doomed the union in all but the Haar case. This is because for all but the Haar wavelet, the supports of the wavelet functions overlap. We will show the connection between the overlapping of the wavelet functions and the internal property and illustrate how the nonoverlapping property of the Haar wavelet permits its simple union with the MAR framework. After proving some particular relationships between wavelet coefficients and through appropriate state augmentation we show how to build internal MAR processes based on any compactly supported wavelet.

An important property that MAR processes possess is widesense Markovianity. This MAR "Markov Property," as we shall call it, is a generalization of the wide-sense Markovianity of state-space processes. For a state-space process, the present is a boundary between the past and future in that it conditionally decorrelates them. Analogously, for an MAR process, the node is a boundary between the subtrees extending away from it; the values of the MAR process indexed by these subtrees are mutually conditionally decorrelated by . The Markov Property means that summarizes all the relevant stochastic properties of one subtree leading from for optimal statistical reasoning about the other subtrees leading from and therefore justifies our terminology in calling an MAR state. For internal MAR processes, the Markov Property is equivalent to the fact that the linear functionals which define the states fulfill the conditional decorrelation role just described. This provides part of our motivation for choosing the linear functionals from a wavelet basis since they have nice decorrelation properties.

Our main objective, after showing how to unify the MAR and wavelet frameworks, is to build approximate internal MAR models for stochastic processes. To do so, we use the statistics of the process to be modeled to derive the dynamics of our internal MAR-wavelet models. While wavelets have nice decorrelation properties, the decorrelation they provide is not exact in general. Therefore, our MAR models based on wavelets are approximate. This does not mean that we assume in our internal models that the detail coefficients are white. In fact, while (for comparison purposes) we do make this assumption for what we shall call the standard MAR-wavelet models, our internal MAR-wavelet models are more sophisticated. In particular, they incorporate the powerful property of optimal stochastic prediction for the detail coefficients at a given scale from both detail and scaling coefficients at coarser scales. This is different from the common wavelet-based modeling in which the detail coefficients are assumed to be white. In our internal models, we make the weaker assumption that the prediction errors are white. This assumption is the reason why our models are approximate.

Another property that an MAR process must posses is low state dimensionality. This is because sample-path generation, linear least squares estimation [START_REF] Chou | Multiscale recursive estimation, data fusion, and regularization[END_REF], [START_REF] Chou | Multiscale systems, Kalman filters, and Riccati equations[END_REF], and likelihood calculation [START_REF] Luettgen | Likelihood calculation for a class of multiscale stochastic models, with application to texture discrimination[END_REF] for MAR processes all have complexity which is polynomial in state dimension (and linear in the number of fine-scale nodes).

We will see that the state dimension of our MAR-wavelet models grows only linearly with the lengths of the support of the scaling functions, which are related, in some cases, such as orthogonal wavelets, to the number of vanishing moments of the analyzing wavelet. However, the fact that wavelets with a large number of vanishing moments do a good job of whitening and stationarizing a large class of processes [START_REF] Abry | The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and fast implementation[END_REF], [START_REF] Flandrin | Wavelet analysis and synthesis of fractional Brownian motion[END_REF], [START_REF] Krim | Multiresolution analysis of a class of nonstationary processes[END_REF], [START_REF] Sellan | Synthèse de mouvements Browniens fractionnaires à l'aide de la transformation par ondelettes[END_REF], [START_REF] Tewfik | Correlation structure of the discrete wavelet coefficients of fractional Brownian motion[END_REF], [START_REF] Zhang | A wavelet-based KL-like expansion for widesense stationary random processes[END_REF], does not imply that the degree of statistical fidelity of our internal models necessarily increases with the number of vanishing moments. This is because we are not exclusively concerned with the correlations between wavelet coefficients, but rather with the conditional correlation between them (in order to approximately satisfy the Markov Property). Therefore, as will be illustrated, with internal MAR-wavelet models, it is possible to build accurate models using wavelets with fairly short supports and, thus, without dramatically increasing the state dimension.

Throughout this paper, we will use fractional Brownian motion (fBm) as a vehicle to support our results. In particular, we will apply our internal MAR-wavelet models to generating fBm sample-paths and, more importantly, to the problem of estimating an fBm from noisy, irregular, and sparse measurements.

In the context of the previous work on MAR processes, this paper provides a unification of wavelets and MAR processes as well as a new view of the multiscale stochastic realization problem. This problem is to design coarse-scale states to match the given fine-scale statistics. Previous approaches to the multiscale stochastic realization problem focussed on designing MAR states to, in some sense, optimally match the statistics of the finest scale process being modeled [START_REF] Frakt | A realization theory for multiscale autoregressive stochastic models[END_REF]- [START_REF] Irving | A canonical correlations approach to multiscale stochastic realization[END_REF]. As a consequence, the resulting states typically have no discernible structure beyond the fact that they represent solutions to specific optimization problems.

Our approach differs in that the design of our MAR states is not closely tied to the intricate details of the fine-scale statistics. Our philosophy which, in part, motivated this work is to restrict the class of linear functionals that define the states of an internal MAR process to the small but rich class of wavelet bases. We thus force the states to contain meaningful multiscale representations of the fine-scale process and avoid the computationally burdensome search over all possible linear functionals.

On the other hand, our approach is similar to the previous work on the multiscale stochastic realization problem in that we use the fine-scale statistics of the process to be modeled to derive the dynamics of the MAR model.

There are at least two other frameworks [START_REF] Chou | A stochastic modeling approach to multiscale signal processing[END_REF], [START_REF] Fabre | New fast smoothers for multiscale systems[END_REF] that propose unifications of multiscale stochastic processes and wavelets, both of which are different from ours in significant ways. To fully contrast [START_REF] Chou | A stochastic modeling approach to multiscale signal processing[END_REF] and [START_REF] Fabre | New fast smoothers for multiscale systems[END_REF] with our work requires a longer discussion than space constraints permit. Therefore, we focus on the main points and refer the reader to [START_REF] Chou | A stochastic modeling approach to multiscale signal processing[END_REF] and [START_REF] Fabre | New fast smoothers for multiscale systems[END_REF] for details. The first [START_REF] Chou | A stochastic modeling approach to multiscale signal processing[END_REF] develops decoupled dynamic wavelet systems that support a fast estimation algorithm whose structure takes the form of a set of monadic trees in scale. The second [START_REF] Fabre | New fast smoothers for multiscale systems[END_REF] generalizes the work in [START_REF] Chou | A stochastic modeling approach to multiscale signal processing[END_REF] to wavelet packets and develops a fast estimation framework whose structure takes the form of a set of dyadic trees. There are several differences between these approaches and ours. The first is that in [START_REF] Chou | A stochastic modeling approach to multiscale signal processing[END_REF] and [START_REF] Fabre | New fast smoothers for multiscale systems[END_REF] it is assumed that the detail coefficients are white. 3 In our internal models, we make no such assumption. The second is that to perform estimation in the frameworks of [START_REF] Chou | A stochastic modeling approach to multiscale signal processing[END_REF] and [START_REF] Fabre | New fast smoothers for multiscale systems[END_REF], the data must be transformed into the wavelet domain. Therefore, there is no way to handle sparse or irregular measurements which our framework can handle easily. Lastly, the models of [START_REF] Chou | A stochastic modeling approach to multiscale signal processing[END_REF], [START_REF] Fabre | New fast smoothers for multiscale systems[END_REF] are only interesting in the noninternal context. Under an internality assumption, they collapse to a trivial case.

There exist other wavelet-based tree models such as those in [START_REF] Cheng | Trainable context model for multiscale segmentation[END_REF], [START_REF] Crouse | Simplified wavelet-domain hidden Markov models using contexts[END_REF], and [START_REF] Crouse | Wavelet-based statistical signal processing using hidden Markov models[END_REF]. In these models, the key wavelet coefficient dependencies are captured using hidden Markov models (HMM's). In particular, these HMM's are used to capture contexts describing statistical features. These models differ from ours in that they include hidden variables and are, therefore, necessarily not internal. Nevertheless, these models also lead to very powerful and efficient algorithms for signal and image processing.

The remainder of this paper is organized as follows. In Section II we review the MAR framework. In Section III we review the essential elements of wavelet theory and introduce the MAR-Haar process. In Section IV we show how to build an internal MAR-wavelet process based on any compactly supported wavelet. In Section V we build approximate MARwavelet models for random signals. The conclusion and a discussion of open questions for future work are in Section VI.

II. MAR PROCESSES BACKGROUND

MAR processes are tree-indexed stochastic processes. For the purposes of this paper it is sufficient to consider only dyadic trees. Our notation for referring to nodes of a dyadic tree is indicated in Fig. 1. There is a natural notion of scale associated with dyadic trees. The root node represents the coarsest scale which we denote as scale zero. The children of the root node represent the first scale. Continuing, leaf nodes constitute the finest scale which we denote as the th scale. We will denote by the stacked vector consisting of for , i.e., the finest scale subprocess. As mentioned, an MAR process has a Markov Property: it is a (wide-sense) Markov random field on a tree [START_REF] Chou | Multiscale recursive estimation, data fusion, and regularization[END_REF]. This Markov Property leads to fast statistical signal processing algorithms. For instance, sample-path generation (with complexity quadratic in state dimension and linear in the number of finestscale nodes) is accomplished using [START_REF] Abry | The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and fast implementation[END_REF]. Also, a linear least squares estimator [START_REF] Chou | Multiscale recursive estimation, data fusion, and regularization[END_REF], [START_REF] Chou | Multiscale systems, Kalman filters, and Riccati equations[END_REF] and likelihood calculator [START_REF] Luettgen | Likelihood calculation for a class of multiscale stochastic models, with application to texture discrimination[END_REF] have been developed based on a measurement model [START_REF] Akaike | Stochastic theory of minimal realizations[END_REF] The zero-mean white noise, has auto-covariance and is uncorrelated with the MAR process 4and the process noise

. The estimator and likelihood calculator have computational complexity cubic in state dimension and linear in the number of finest-scale nodes.

An important subclass of the class of MAR processes are internal MAR processes. As stated in Section I, an internal MAR process is one for which each state is a linear function of the portion of indexed by finest scale nodes which descend from . Therefore, if is an internal MAR process we can write [START_REF]Markovian representation of stochastic processes by canonical variables[END_REF] We will call an internal matrix. As shown in [START_REF] Frakt | A realization theory for multiscale autoregressive stochastic models[END_REF] and [START_REF] Frakt | Computationally Efficient Stochastic Realization for Internal Multiscale Autoregressive Models[END_REF], a necessary and sufficient condition to fulfill ( 3) is for each state to be a linear function of and , the states which reside at the children nodes of . Notice that this fact imposes some constraints on the form of and thus on . The internal matrices cannot be arbitrarily and independently defined but are coupled so as to arrive at an internal process [START_REF] Frakt | A realization theory for multiscale autoregressive stochastic models[END_REF], [START_REF] Frakt | Computationally Efficient Stochastic Realization for Internal Multiscale Autoregressive Models[END_REF].

The concept of an internal process was first introduced in the state space time-series modeling literature [START_REF] Akaike | Stochastic theory of minimal realizations[END_REF]- [START_REF] Arun | Balanced approximation of stochastic systems[END_REF], [START_REF] Desai | A realization approach to stochastic model reduction and balanced stochastic realizations[END_REF]. In the time-series modeling context, internal models are of intense interest because they constitute a class of processes which is rich enough to include minimal models (i.e., those with the smallest possible state dimensions). While this is not the case for internal MAR models (i.e., a minimal MAR model may not belong to the class of internal models), there are, nonetheless, several good reasons why we are interested in internal MAR processes. As mentioned in Section I, internal processes are of both intellectual and practical interest. First, the theory for internal MAR processes is well-developed and provides the firm foundation upon which our work is built. Indeed, many of the concepts relating to internal processes developed in the state-space context have been generalized to MAR processes in [START_REF] Frakt | A realization theory for multiscale autoregressive stochastic models[END_REF]- [START_REF] Irving | A canonical correlations approach to multiscale stochastic realization[END_REF].

Second, while noninternal MAR processes can be constructed, their states have exogenous random components, a property that is not suitable in many problems. In contrast, internal processes consist of states which are linearly related, a property that is essential to address certain problems, for instance when one wishes to estimate coarse-scale states which are, in fact, local averages of a fine-scale signal. In addition, with internality we can make clear the tie of the MAR framework to wavelet analysis.

Finally, internal processes provide a convenient parameterization for the MAR dynamics. From the internal matrices, the dynamics are related in a simple way to the covariance of the fine-scale process 5. The key to seeing this is recognizing that (1) represents the linear least squares estimate of from plus the estimation error . Therefore, the standard linear least squares formulas [START_REF] Van Trees | Detection, Estimation, and Modulation Theory, Part I[END_REF] dictate that (4) [START_REF] Basseville | Modeling and estimation of multiresolution stochastic processes[END_REF] where is our notation for the covariance matrix for random vector and is the cross covariance matrix for random vectors and . The state covariances and cross-covariances follow trivially from (3) as (6) [START_REF] Chou | Multiscale recursive estimation, data fusion, and regularization[END_REF] While a full theoretical development of internal MAR processes may be found in [START_REF] Frakt | Computationally Efficient Stochastic Realization for Internal Multiscale Autoregressive Models[END_REF], we point out one aspect of this theory so as to avoid confusion. An MAR process has white noise-driven coarse-to-fine dynamics [cf. [START_REF] Abry | The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and fast implementation[END_REF]]. Yet, for an internal MAR process, the parent state is a linear function of its children states, a deterministic fineto-coarse relationship. That is, for some matrix we have . That the driving noise and are uncorrelated and that is deterministically related to and is not contradictory because by construction [i.e., using (4)-( 7)], the noise vector is in the null-space of [START_REF] Frakt | Computationally Efficient Stochastic Realization for Internal Multiscale Autoregressive Models[END_REF]. In this paper, our primary interest is the multiscale stochastic realization problem (hereafter called simply the realization problem). The realization problem is to build an MAR process such that the fine-scale realized covariance matrix, , matches a specific desired covariance matrix which we will denote by . Of particular interest to us is the decorrelation role that the linear functionals [the rows of ] play in the realization problem. To achieve , the linear functionals which define the states of the MAR process must represent enough information to satisfy the Markov Property. However, for many problems this requires impractically large state dimensions, resulting in models which are not useful because of the high computational burden of sample-path generation, estimation, or likelihood calculation.

There are several ways to overcome this computational bottleneck. One way is to start with an exact model and then to reduce the state dimensions by throwing away some of the linear functionals which define the states [START_REF] Frakt | A realization theory for multiscale autoregressive stochastic models[END_REF]- [START_REF] Irving | A canonical correlations approach to multiscale stochastic realization[END_REF]. One can then compute the dynamics of an MAR model by taking the resulting reduced row-dimension internal matrices and using them in ( 4)- [START_REF] Chou | Multiscale recursive estimation, data fusion, and regularization[END_REF] in which is substituted for . A consequence of reducing the state dimensions is that the Markov Property becomes an approximation and the resulting model is not exact . For exact internal models, throwing away elements of states will, in general, also destroy internality. This will occur if, for any node , the state cannot be written as a linear function of its children states because the necessary information has been discarded in the process of state dimension reduction. This raises the following interesting question: how can low-order approximate models (i.e., those for which

) also be made internal? Our approach to this question, in particular, and to the stochastic realization problem, in general, differs considerably from previous work [START_REF] Frakt | A realization theory for multiscale autoregressive stochastic models[END_REF]- [START_REF] Irving | A canonical correlations approach to multiscale stochastic realization[END_REF] in several ways. In this previous work, the states of exact models are closely tied to the process being modeled in that the internal matrices are determined by solving optimization problems, the parameters of which are governed by the intricacies of fine-scale desired covariance . Therefore, these internal matrices have no meaningful structure beyond the fact that they are solutions to specific optimization problems.

In contrast to these previous approaches, our internal matrices are not the solutions to optimization problems. Instead, they are selected from the small but rich class of wavelet bases. Therefore, they take no work to compute and have the intuitively pleasing and mathematically rich structure given by wavelets. Moreover, we directly design approximate internal models without first constructing an exact model and then discarding state information.

III. WAVELET BACKGROUND

In this section, we give a brief review of wavelet decompositions. For more details see [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF], [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF], [START_REF] Meyer | I: Ondeletts, II: Opérateurs de Calderón-Zygmund, III: Opérateurs Multilinéaires[END_REF], and [START_REF] Strang | Wavelets and Filter Banks[END_REF]. The wavelet representation of a continuous signal consists of a sequence of approximations of at coarser and coarser scales where the approximation at the th scale consists of a weighted sum of shifted and dilated versions of a basic function called the scaling function. By considering the incremental details added in obtaining the st scale approximation from the th, one arrives at the wavelet transform based on a single function called the analyzing wavelet.

The reconstruction is performed using the function , called the synthesis wavelet, such that the two families and are a biorthogonal Riesz basis of where and similarly for . The synthesis wavelet is obtained from the function which is dual to , i.e., which satisfies , where is the standard inner product in and is the Dirac function. The scaling functions and must satisfy and where and are discrete filters satisfying the biorthogonality condition in [START_REF] Chou | Multiscale systems, Kalman filters, and Riccati equations[END_REF] and are given by where (9)

The discrete filters and must satisfy the perfect reconstruction condition which can be found in [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]. When and , then is a conjugate mirror filter (CMF) and the family constitutes an orthonormal wavelet basis of . The fast wavelet transform computes the wavelet coefficients of a discrete signal . The fast wavelet decomposition algorithm is (10) Fig. 2. The Haar dependency graph is a dyadic tree. Here, n is even.

The reconstruction algorithm is [START_REF] Cohen | Wavelet bases on the interval and fast algorithms[END_REF] The coefficients and are called, respectively, the scaling and detail coefficients at the th scale and th shift.

In the remainder of this paper, we consider only the case when and are finite impulse response (FIR) filters, i.e., when they have a finite number of nonzero coefficients. For the sake of notational simplicity, we assume that the lengths of and are both even. Without loss of generality, we choose and for some integers and such that . Thus, using ( 9) we have and . We also assume that and have the same parity. We point out, however, that all the results in this paper hold for all perfect reconstruction FIR filters with minor modifications.

The wavelet reconstruction algorithm (11) defines a dynamical relationship between the scaling coefficients at one scale and those at the next finer scale, with the detail coefficients acting as the input. Note that these dynamics are with respect to scale rather than time. This suggests that it is natural to think of constructing MAR processes within the wavelet framework. This construction is, in fact, obvious in the case of the Haar wavelet [START_REF] Fieguth | Fractal estimation using models on multiscale trees[END_REF] because the dependency structure of the wavelet coefficients is a dyadic tree as shown in Fig. 2. Indeed, the wavelet reconstruction algorithm using the Haar system states that, for each [START_REF] Crouse | Simplified wavelet-domain hidden Markov models using contexts[END_REF] Assuming that the detail coefficients are white noise, [START_REF] Crouse | Simplified wavelet-domain hidden Markov models using contexts[END_REF] suggests that one can build an MAR process [START_REF] Fieguth | Fractal estimation using models on multiscale trees[END_REF] by defining each as containing a scaling and detail coefficient at scale and position , i.e.,

. For , the auto-regression for this model is [START_REF] Crouse | Wavelet-based statistical signal processing using hidden Markov models[END_REF] where represents . The auto-regression at the finest scale is simply given by [START_REF] Daniel | Modeling and estimation of fractional Brownian motion using multiresolution stochastic processes[END_REF] The link between MAR processes and wavelets is not obvious if one considers wavelets other than the Haar system. This is due to the overlapping supports of such wavelets (which does not occur in the Haar case). Indeed, to compute a scaling coefficient one does not need only the time-synchronous parents 6 of but also a number of neighboring coefficients depending on the supports of the analysis and synthesis wavelet. Thus, if we build a multiscale process where the states are defined as in the Haar case, i.e., , but where we consider that the scaling and detail coefficients are computed using more regular wavelets, we will end up with a more complex graph structure of the scale-to-scale autoregression instead of a tree one as imposed by [START_REF] Abry | The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and fast implementation[END_REF]. This is illustrated in Fig. 3 in the case of the Daubechies four-tap filter [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF].

The first issue, then, is to redefine the states in such a way as to arrive at a tree dependency structure rather than a more complex graph. We will see that this can be done easily using state augmentation. The second issue we must address, which is the crucial issue and the most difficult one, is how to provide internality. As mentioned in Section I, this is, in fact, one of our main objectives since we want to use wavelets as the linear functionals that define the internal states. These two issues will be the focus of the next section.

IV. WAVELET-BASED INTERNAL MAR PROCESSES

In this section, we first address (in subsection A) the issue of defining the states in such way to obtain a tree dependency structure. We then address (in subsection B) the issue of internality.

A. Tree Structure

To see the intuition behind how to define the states in order to arrive at a tree structure, let us consider the simple case where is the Daubechies four-tap filter [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF]. In this case, we have . Then, the wavelet reconstruction algorithm [START_REF] Cohen | Wavelet bases on the interval and fast algorithms[END_REF] implies that for every even integer [START_REF]A multiresolution methodology for signal-level fusion and data assimilation with applications to remote sensing[END_REF] Therefore, for every and for every , if we chose each state to be it is clear from (15) that the scaling coefficients carried within each depend only on the parent of (see Fig. 4).

In the general case, for every and for every , the state at scale and shift is defined as shown in [START_REF] Daoudi | Multiscale autoregressive models and wavelets: Extended version[END_REF], at the bottom of this page. The details showing that [START_REF] Daoudi | Multiscale autoregressive models and wavelets: Extended version[END_REF] implies that each state depends only on its parent can be found in Appendix A. We then can show that if if [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF] if if [START_REF] Daoudi | Multiscale autoregressive models and wavelets: Extended version[END_REF] Authorized licensed use limited to: UR Futurs. Downloaded on December 4, 2009 at 12:37 from IEEE Xplore. Restrictions apply. where [START_REF] Daubechies | Factoring wavelet transforms into lifting steps[END_REF] The proof of ( 17) and the expression for the matrices can also be found in Appendix A.

Assuming that is a white noise process uncorrelated with the root node state , ( 17) represents an MAR process with dynamics matching the reconstruction algorithm associated with any compactly supported orthogonal or biorthogonal wavelet. In the sequel, we refer to this process as the standard MAR-wavelet process.

B. The Internal MAR-Wavelet Process

If the coefficients are considered as the detail coefficients computed using the wavelet decomposition algorithm and thus are deterministic inputs, then (17), ( 18) is just a rewriting of the wavelet reconstruction algorithm [START_REF] Cohen | Wavelet bases on the interval and fast algorithms[END_REF]. But if the coefficients are generated as random variables, then ( 17) and ( 18) constitute a statistical model for a finescale process. However, almost surely, the states generated by this model do not consist of scaling and detail coefficients 7 of the realized fine-scale process . This is because the standard MAR-wavelet process is not internal. Indeed, as mentioned in Section II, a necessary and sufficient condition for internality is that each state has to be a linear functional of its immediate children [START_REF] Frakt | A realization theory for multiscale autoregressive stochastic models[END_REF]- [START_REF]Efficient multiscale stochastic realization[END_REF]. From the wavelet decomposition algorithm, one can easily see that, when , each state of the standard MAR-wavelet process is not a linear functional of just its immediate children but of the range of states at the next finer scale depending on the supports of the scaling functions.

The question now is how to build an internal MAR-wavelet process in order to ensure that the states consist of scaling and detail coefficients of the realized fine-scale process. This issue is, in fact, the one which seemed to doom the union between MAR process and wavelets, and the central contribution of this paper is to solve this problem. This will be done by exhibiting and exploiting some relationships between scaling and detail coefficients and by appropriately modifying the state definition. We emphasize that this is purely deterministic analysis.

The intuition that leads us to construct internal states is essentially state augmentation. For the coarse-to-fine synthesis, each state needs to have all the necessary information. This was done in the standard MAR-wavelet process. We need now to make sure that each state has all the information for analysis, so that we have internality. As explained above, the states of the standard MAR-wavelet process do not contain enough information. We thus need to augment the states in order to achieve internality. Before showing how we augment the states, we need the following intermediate result which allows us to add only a few coefficients in the process of defining internal states. The idea behind constructing internal states is to define new states in such a way so that the left (respectively, right) child of contains (respecticely, However, having copied and from to its children and , we must continue to pass and down to the children (and grandchildren and so on) of and to maintain internality. Of course, we must do this for all and . This seems to suggest that the state dimensions will explode. However, by simply splitting at each step the necessary information between the two children, the state dimension remains bounded. The construction of the states is depicted in Fig. 5 in the simple case of Daubechies four-tap filter. To define rigorously the internal states in the general case, for , we define recursively the sequence of vectors as shown in ( 21) at the bottom of this page. We then have the final result in which we show in bold the augmentation of the states with respect to the standard MAR-wavelet model. This is shown in [START_REF] Flandrin | Wavelet analysis and synthesis of fractional Brownian motion[END_REF] Proposition 2: The MAR process for which the states are defined by if is even if is odd [START_REF] Fieguth | Fractal estimation using models on multiscale trees[END_REF] Authorized licensed use limited to: UR Futurs. Downloaded on December 4, 2009 at 12:37 from IEEE Xplore. Restrictions apply. [START_REF] Flandrin | Wavelet analysis and synthesis of fractional Brownian motion[END_REF] is internal.

Proof: See Appendix C.

We refer to this new process as the internal MAR-wavelet process. Notice that the size of each is . Thus the maximal state dimension of the internal MAR-wavelet process is . With Proposition 2, we have shown how to build internal MAR processes based on any compactly supported orthogonal or biorthogonal wavelet. This completes our unification of wavelets with MAR processes.

V. APPROXIMATE MAR-WAVELET MODELS FOR STOCHASTIC PROCESSES

In this section, we focus on the construction of approximate MAR-wavelet models for stochastic processes in order to take advantage of the signal processing algorithms associated with the MAR framework. In our examples, we will use the fast sample-path generator and the fast linear least squares estimator for estimating a signal from irregularly spaced, sparse measurements corrupted by nonstationary noise.

The standard MAR-wavelet process, defined by ( 17) and ( 18), can be used as an approximate model for a stochastic process by assuming that the detail coefficients are white. We call this model the standard MAR-wavelet model. However, the states realized using this model are not consistent with the finescale realized process in that they do not represent, with probability one, scaling and detail coefficients of . This is because of the lack of internality, as discussed in Section IV-B. Notice that the assumption of the whiteness of [defined by [START_REF] Daubechies | Factoring wavelet transforms into lifting steps[END_REF]] is never fulfilled if . Indeed, for and such that , it is clear that, at a given scale and are correlated since they share at least one detail coefficient.

By achieving internality, the states of the internal MARwavelet process [START_REF] Flandrin | Wavelet analysis and synthesis of fractional Brownian motion[END_REF] are forced to be consistent with the fine-scale realized process. We can then examine the problem of building internal stochastic models that are consistent with the graph structure of the tree (i.e., we insist on models in which we have the Markov Property) and approximate the given statistics of a fine-scale process. Given these fine-scale statistics, internality provides immediately the statistics of any MAR state and the statistics between each state and its parent, as each state is a set of linear functionals of the fine scale. As a result, we can immediately define the linear dynamics of an MAR model of the form [START_REF] Abry | The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and fast implementation[END_REF]. The dynamics of this model incorporate optimal prediction from parent to child using the implied statistics, and we then model the errors in this prediction as white (the requirement for the Markov Property to be satisfied).

The implications of this are twofold. First, the resulting internal model in general produces fine-scale statistics that only approximate the desired ones (because of our insistence that the coarse-to-fine prediction errors be white). To be sure, our internal MAR-wavelet model does produce the correct marginal statistics at each node and the correct joint statistics for each state and its parent, but other statistics (e.g., cross-covariance for two nodes at the same scale) are only captured approximately. The second point is that the coarse-to-fine dynamics so defined are in general very different from standard wavelet modeling. In particular, these dynamics exploit correlation between detail coefficients and coarser scale scaling and detail coefficients by performing optimal prediction and then assuming that the errors in these predictions are white. This is in contrast to one common approach in using wavelets for modeling stochastic processes in which the detail coefficients are themselves modeled as white (i.e., the wavelet representation is assumed to be the Karhunen-Loeve (K-L) decomposition). In our case, since we allow MAR dynamics, we do not need to have K-L diagonalization. Rather, the success of our method in approximating stochastic processes relies only on the weaker requirement that the errors in predicting finer scale detail coefficients from coarser scale coefficients are white. As we illustrate, an implication of this is that we can use fairly short wavelets, implying lower state dimensions, which certainly do not do a good job of whitening the details (as evidenced by our results using the noninternal standard MAR-wavelet models), but which do remarkably well for our internal models.

Using this optimal prediction procedure, we incorporate a synthesis algorithm for the detail coefficients in addition to the usual wavelet reconstruction algorithm for the scaling coefficients. The initialization for this new synthesis algorithm is given by the statistics of the scaling and detail coefficients at the coarsest scale. Those statistics are given by the covariance matrix of the root node state which is computed using (6) (in which is replaced by ). More precisely, the optimal prediction is performed as follows: if represents the detail coefficients carried by the state defined by [START_REF] Flandrin | Wavelet analysis and synthesis of fractional Brownian motion[END_REF], then [START_REF] Frakt | A realization theory for multiscale autoregressive stochastic models[END_REF] where the covariance matrix for is and where , and are submatrices of and which are computed using ( 6) and [START_REF] Chou | Multiscale recursive estimation, data fusion, and regularization[END_REF] in which is replaced by , the covariance matrix for the signal being modeled. 8 The prediction errors are not white in general. This can be easily seen from the fact that the states of the internal MAR-wavelet model contain duplicated detail coefficients. Yet, we assume that the prediction errors in [START_REF] Flandrin | Wavelet analysis and synthesis of fractional Brownian motion[END_REF] are white noise to legitimately apply the signal processing algorithms provided by the MAR framework. We thus have an approximate MAR model for any process whose second-order statistics, , are known. We will refer to this model as the internal MAR-wavelet model. Note that an advantage of the internal MAR-wavelet model and, in fact, of any internal model is that it achieves the correct variances (i.e., the diagonal elements of match exactly those of ). We important point. In real world problems, the user may not know how to choose the appropriate wavelet to do a good job in decorrelating the process under study. Thus, the resulting detail and scaling coefficients may be strongly correlated. We point out that, even in this case, our internal MAR-wavelet model can be an accurate one for the underlying process by exploiting these potential correlations, as well as the correlations between detail and scaling coefficients, in the optimal prediction scheme. We will illustrate this later in the case of an fBm.

To be more precise, let us discuss this optimal prediction algorithm in the simple case of the Haar wavelet and support this discussion by showing some examples for fBm. We will consider fBm defined on the time interval and we normalize the fBm statistics to have unit variance at time one. The covariance function for fBm [START_REF] Mandelbrot | Fractional Brownian motions, fractional noises and applications[END_REF] is [START_REF] Frakt | Computationally Efficient Stochastic Realization for Internal Multiscale Autoregressive Models[END_REF] Fig. 6(a) and (b) shows for 64 samples of fBm on the interval with Hurst parameter and , respectively.

A. The Haar Case

The MAR-Haar model defined in Section III by ( 13) and ( 14) has many drawbacks. First, the assumption that the detail coefficients are white is very poor in general. Indeed, since the Haar wavelet has only one vanishing moment, for most processes the resulting detail coefficients are strongly correlated both in space and scale. Second, due to the piecewise constant shape of the Haar wavelet, any realized covariance matrix (i.e., the covariance matrix of ) with this model will have "blockiness" in general. Thus, any sample-path generated using this model will have distracting artifacts and will be blocky. Therefore, this model is not appropriate for synthesizing or estimating stochastic processes in general. An illustration of this phenomenon is shown in Fig. 7(a) which displays for an fBm with Hurst parameter . It is clear by comparing Fig. 6(a) and Fig. 7(a) that is a poor approximation to . We point out, however, that this model has been successfully used for hypothesis discrimination. In [START_REF] Fieguth | Fractal estimation using models on multiscale trees[END_REF] the authors applied the MAR likelihood calculator to accurately estimate the Hurst parameter of fBm. Now, notice that this model is internal, which is clear from the wavelet decomposition algorithm associated with the Haar system. Therefore, instead of using the dynamics defined by ( 13) and ( 14), one can build a more accurate model [START_REF] Daniel | Modeling and estimation of fractional Brownian motion using multiresolution stochastic processes[END_REF] by computing the auto-regression parameters so that the multiscale auto-regression is the optimal prediction of from . Thus, while in the model defined by ( 13) and ( 14) represents the detail coefficient , the process noise in the internal model based on optimal prediction will represent the prediction error in the estimation of conditioned on the detail and scaling coefficient represented by , i.e., where, from the linear least squares estimation error formula, we have

Authorized licensed use limited to: UR Futurs. Downloaded on December 4, 2009 at 12:37 from IEEE Xplore. Restrictions apply.

(a) (b) (c) Fig. 7. Realized covariance matrices P J for fBm with H = 0:3. (a) P J using the standard MAR-Haar model. (b) P J using the internal MAR-Haar model. (c) jP f 0 P J j where P J is from (b).

Therefore, this multiscale model will capture correlations in scale among the detail coefficients represented by states at neighboring nodes in the tree. Thus, this optimal prediction model will do a better job of approximating the statistics of the underlying process than does the model defined by ( 13) and ( 14). The improvement is illustrated in the case of fBm in Fig. 7(b) which displays the realized covariance matrix of an fBm with Hurst parameter and should be compared to Fig. 7(a). This shows the power of the optimal prediction procedure in the simple case of the Haar wavelet.

B. The General Case

It is worth noting that, with the Haar wavelet, the detail coefficients which are not neighbors (in space and scale) are in general also strongly correlated. Therefore, even with this optimal prediction procedure, the internal MAR-Haar model remains very crude since it captures only the correlations between a detail coefficient at a given scale and the timesynchronous detail and scaling coefficients at the previous coarser scale. Fig. 7(b) and (c) illustrates the limitation of this model in the case of fBm with . One sees that the realized covariance matrix is still a poor approximation of the true one.

One way to overcome the limitations of the Haar wavelet is to build an internal MAR-wavelet model using an analyzing wavelet with a large number of vanishing moments. With such a wavelet, the detail coefficients which are not neighbors in space and scale will, in general, be better decorrelated and the potential correlations will reside only between neighboring coefficients. Then, our optimal prediction procedure will exploit these residual correlations between detail and scaling coefficients and do the best job in linearly predicting the detail coefficients.

However, this is not the only solution. One can still build accurate models without using an analyzing wavelet with large number of vanishing moments. Indeed, all we need to have accurate models is to provide a good approximation to the Markov Property. Therefore, accurate models will be provided using any wavelet yielding scaling and detail coefficients such that the states they form approximatively fulfill the conditional decorrelation role of the Markov Property.

To support these arguments and to illustrate the performance of the optimal prediction procedure, we apply our MARwavelet models to approximate the statistics of fBm using different wavelets. Specifically, we compare the internal MARwavelet model to the standard MAR-wavelet model which assumes the whiteness of the detail coefficients. We use the Daubechies orthogonal wavelet with two vanishing moments (Daub4), the Daubechies orthogonal wavelet with three vanishing moments (Daub6), the spline biorthogonal wavelet (Spline13) such that 9[respectively, ] has three (respectively, 1) zeros at , and the spline biorthogonal wavelet (Spline31) such that [respectively, ] has 1 (respectively, three) zeros at . Fig. 8(a)-(c) displays the element-wise absolute value of the difference between and obtained by the standard MAR-wavelet model for an fBm with using, respectively, Daub4, Daub6, and Spline13. The improvement with respect to the standard MAR-Haar model is clear as expected, since we are using analyzing wavelets with more than one vanishing moment. However, the approximation is not satisfactory which is not surprising since the detail coefficients are not exactly decorrelated using these wavelets. Finally, note that Daub6 does better than Spline13 because Daub6 is an orthogonal wavelet and is smoother than the analyzing wavelet of Spline13. Now, with the internal MAR-wavelet model, the detail coefficients are no longer assumed to be white noise. Instead, they are computed using the optimal prediction procedure described above. Therefore, the internal MAR-wavelet model will better approximate the statistics of fBm. Fig. 9(a)-(c) displays the element-wise absolute value of the difference between and for an fBm with using, respectively, Daub4, Daub6, and Spline13. The improvement with respect to the standard MAR-wavelet model is clear. of the difference obtained using the internal MAR-wavelet model for fBm with . To illustrate, in the case of fBm, the fact that even with relatively nonregular wavelets our internal MAR-wavelet model can provide very accurate models, we use the biorthogonal wavelet Spline31. The analyzing wavelet for Spline31 has only one vanishing moment and the synthesis wavelet is extremely singular (see Fig. 11). Fig. 12(a) displays the element-wise absolute value of the difference between and using the standard MAR-wavelet model. One sees that the approximation is extremely bad, which is not surprising given the properties of Spline31 and the weakness of the assumption that the detail coefficients are white. However, using the internal MAR-wavelet model, the approximation is very accurate as displayed in Fig. 12(b). Furthermore, notice that this approximation is more accurate than the one illustrated in Fig. 10(c) in which the state dimension is larger. Indeed, in Fig. 12 maximum state dimension is eight while in Fig. 10(c) we have and thus the maximum state dimension is ten. This shows the power of the optimal prediction procedure in approximating the Markov Property even without considering analyzing wavelets with a large number of vanishing moments. Now, we use the fast signal processing algorithms associated with the MAR framework to synthesize fBm sample-paths and to perform estimation from irregularly-spaced and sparse measurements corrupted by nonstationary noise. We emphasize that the latter is a problem which cannot be easily and efficiently handled with other estimation techniques due to the nonstationarity of the process to be estimated and the process noise and the irregularity of the measurements. Fig. 13 (over which the white measurement noise has 10 Exact realizations of fBm are obtained by multiplying white Gaussian noise by the matrix square root of P f . This requires O(N 3 ) computations if P f is N 2 N . In contrast, the MAR sample-path generator is O(N ). variance 0.5). Fig. 14(c) displays the MAR estimates based on 14(b) using the internal wavelet model with Daub6. The MAR estimates are the solid line and the optimal estimates based on the exact statistics 11 are the dash-dot line. The plus/minus one standard deviation error bars are the dashed line. Fig. 15 illustrates the same processing but for fBm with . Notice that in both Fig. 14(c) and in Fig. 15(c) the optimal estimate based on the exact statistics is not easily distinguishable from the MAR estimate since the two nearly coincide. Also, the estimation error standard deviations that the MAR estimator provides are very close to the ones based on the exact statistics (although we have not plotted the latter in our examples). More importantly, the difference between the optimal estimate and the MAR estimate is well within the one standard deviation error bars. This demonstrates that the degree to which our internal MAR-wavelet model deviates from the exact model is statistically irrelevant.

Finally, we point out that in the case of finite length signals, folded discrete transforms [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF] can be applied in the case of symmetric or antisymmetric biorthogonal wavelets. Boundary wavelets [START_REF] Cohen | Wavelet bases on the interval and fast algorithms[END_REF] can be used in the case of orthogonal wavelets. We used the folded discrete transform in both cases. In all the examples we considered, the results were very close to those obtained by assuming the knowledge of the statistics of the signal over . However, at this point, we have no precise mathematical description for the influence of the boarder treatment techniques on the accuracy of the models.

VI. CONCLUSION

The primary contribution of this paper has been to provide a unification of the MAR framework with the wavelet framework. We have shown how to construct internal MAR processes based on any compactly supported orthogonal or biorthogonal wavelet. We then used these internal MARwavelet processes as approximate models for stochastic processes. The marriage of the MAR and wavelet frameworks led us to incorporate a powerful reconstruction algorithm for the detail coefficients which complements the usual wavelet reconstruction algorithm for the scaling coefficients. While we have used fBm as a vehicle to illustrate the performance of our internal MAR-wavelet models, these models can be applied to any process whose second-order statistics exist. While, in this paper, we have assumed that these second-order statistics are known (i.e., is known), this is not a prerequisite: the internal MAR-wavelet dynamics can be efficiently estimated directly from data.

In this way, our approach to modeling represents a new view of the multiscale stochastic realization problem. Rather than selecting internal matrices which are finely tuned according to the particular signal being modeled (as previous approaches do), we select them from a library of linear functionals. Our approach is not only computationally fast, but it provides a systematic way to build MAR states that have meaningful and useful structure and are not tied directly to the intricate details of the process being modeled.

The idea of selecting internal matrices from a library of linear functionals is an extremely important one in many applications. For example, if one is interested in estimating nonlocal variables from data collected at multiple resolutions, the MAR states must be able to represent the coarser variables in addition to playing a conditional decorrelation role. This corresponds to selecting internal matrices from a library of linear functionals which include ones that not only do a good conditional decorrelation job but also include ones which represent nonlocal variables to estimate or to include as measurements. The particular library consisting of wavelet bases represents a natural candidate for this role and might be useful for data fusion problems in which data are available at several resolutions.

While we have shown how to build MAR processes for modeling one-dimensional signals, our approach generalizes to higher dimensions. In two dimensions, there are a number of important and challenging problems associated with the modeling and estimation of random fields. We are currently working on extending the ideas in this paper to applications in image processing.

In addition to extending our work to two-dimensional problems, there are a number of other interesting and challenging problems to consider for future research. First, having states which contain wavelet coefficients suggests ways of doing data-adaptive multiresolution estimation where the resolution of the estimate is governed by the quality of the data. Since the MAR estimator provides estimates of for all , one can use the estimates of the coarser-scale scaling coefficients of an MAR-wavelet model to obtain a coarser-scale signal estimate. Generalizing this idea, one can consider estimating the signal at a fine scale in areas where the data are of relatively high quantity and quality and at a coarser scale where the data are more sparse and of poor quality.

Another challenging problem is to build MAR-wavelet models from incomplete covariance information (i.e., from just parts of ). Solutions to problems of this type have been elusive because, before our work, the linear functionals which define the states were based on complete knowledge of and the relaxation of this has proven difficult. However, since we select linear functionals from a library, we have no need for knowledge of until we want to determine the MARwavelet dynamics. However, at a given node , the MAR parameters and do not rely on all of since they are a function just of and . These are small matrices and, therefore, the dynamics which rely on unavailable parts of can be efficiently estimated from data (if available) or chosen to be consistent with the pieces of which are known.

Finally, there are a number of open questions associated with the work presented in this paper. One is how to choose the right wavelet which would provide the most accurate model for a particular class of processes. While there are many ways to address this issue, the one which is of most interest to us is to use the lifting scheme concept [START_REF] Claypoole | Adaptive wavelet transforms via lifting[END_REF], [START_REF] Daubechies | Factoring wavelet transforms into lifting steps[END_REF] to perform multiscale decompositions which are adapted to the statistics of the process under study. Another interesting question is: how does modeling performance translate to performance in a particular estimation problem? These are questions which will motivate our future work on MAR-wavelet models.

APPENDIX A PROOF OF [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF] To see that [START_REF] Daoudi | Multiscale autoregressive models and wavelets: Extended version[END_REF] implies that each state depends only on its parent, consider two states and at scale , for some even integer . The parent of these two states is Then, for every integer , we have [START_REF]Efficient multiscale stochastic realization[END_REF] and [START_REF] Irving | A theory for multiscale stochastic realization[END_REF] In order to check that every and in ( 25) and ( 26) is carried by , one can easily check that and that

Then, using ( 25) and ( 26), we get [START_REF] Daubechies | Ten Lectures on Wavelets[END_REF] where, for every , the matrices are

Fig. 3 .

 3 Fig. 3. Dependency graph for the Daubechies four-tap filter. Here n is even.

Fig. 4 .

 4 Fig. 4. Through state augmentation, the dependency graph for the Daubechies four-tap filter can be made into a tree. Here n is even.

Fig. 5 .

 5 Fig. 5. Example of the internal MAR-wavelet process with the Daubechies four-tap filter. Scaling coefficients in bold illustrate the necessary information transmitted from one scale to the next. The boxed coefficients are a linear function of the coefficients of their children by the wavelet decomposition algorithm.

Proposition 1 :

 1 There exists four matrices such

Fig. 6 .

 6 Fig. 6. Exact covariance matrices for fBm. (a) H = 0:3. (b) H = 0:7.

Fig. 10 (Fig. 8 .Fig. 9 .Fig. 10 .

 108910 Fig. 8. jP f 0 P J j for fBm with H = 0:3 using the standard MAR-wavelet model. (a) Daub4 (state dimension six). (b) Daub6 (state dimension ten). (c) Spline13 (state dimension eight).

  (c) Spline13 (state dimension ten).

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Spline biorthogonal wavelet (Spline31). (a) Analyzing wavelet. (b) Synthesis wavelet.

  (a) and (b) displays 256-point sample-paths using the internal MARwavelet model with Daub6 for an fBm with and , respectively. Fig. 14(a) displays an exact 64-point realization 10 of fBm with . Fig. 14(b) displays noisy observations of (a) where observations are only available on (over which the white measurement noise has variance 0.3) and

Fig. 13 .Fig. 14 .Fig. 15 .

 131415 Fig. 13. fBm sample-paths using the internal MAR-wavelet model with Daub6. (a) H = 0:3. (b) H = 0:7.

In this paper, we will assume without loss of generality that all processes we consider are zero-mean.

We will use the notation (j; n) to refer to the node of a dyadic tree at scale j and shift n.

Authorized licensed use limited to: UR Futurs. Downloaded on December 4, 2009 at 12:37 from IEEE Xplore. Restrictions apply.

Moreover, the approaches in[START_REF] Chou | A stochastic modeling approach to multiscale signal processing[END_REF] and[START_REF] Fabre | New fast smoothers for multiscale systems[END_REF] assume a particular model while we construct one.

We will use the notation x(1) and w(1) to refer to x j (n) and w j (n), respectively, for all j and n.

E[START_REF] Abry | The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and fast implementation[END_REF] is the expectation operator.

The time-synchronous parents of a j (n) [or d j (n)] are a j01 ([n=2]) and d j01 ([n=2]).

For each state x j (n), only a j (n) and d j (n) can be expressed as linear functions of the children states. Authorized licensed use limited to: UR Futurs. Downloaded on December 4, 2009 at 12:37 from IEEE Xplore. Restrictions apply.

The internal matrices required in (6) and (7) are implicitly given by Proposition 2.

H (z) (respectively, H(z)) is the z-transform of h(n) (respectively, h(n)).Authorized licensed use limited to: UR Futurs. Downloaded on December 4, 2009 at 12:37 from IEEE Xplore. Restrictions apply.

That is, the optimal estimates are obtained by solving the normal equations based on the true fBm and measurement statistics. Note that solving the normal equations requires O(N 3 ) computations while the MAR estimator is O(N ) where N is the size of the signal to be estimated[START_REF] Chou | Multiscale recursive estimation, data fusion, and regularization[END_REF]-[START_REF] Chou | A stochastic modeling approach to multiscale signal processing[END_REF].

. . . . . . . . . (31) for and otherwise for and otherwise for and otherwise for and otherwise Authorized licensed use limited to: UR Futurs. Downloaded on 4, 2009 at 12:37 from IEEE Xplore. Restrictions apply.

ACKNOWLEDGMENT

The authors would like to thank the reviewers for their useful and insightful comments which helped to vastly improve the presentation of the paper.

This work was supported by the Office of Naval Research under Grant N00014-91-J-1004, the Air Force Office of Scientific Research under Grant F49620-98-1-0349, and by INRIA Rocquencourt, France.

and follow from [START_REF]Efficient multiscale stochastic realization[END_REF] and [START_REF] Irving | A theory for multiscale stochastic realization[END_REF] as: for , for and for When are vectors of length and follow as: for and for APPENDIX B PROOF OF PROPOSITION 1 In this Appendix we provide a proof of Proposition 1. The proof of Proposition 1 requires the following lemma. Due to the lack of space, the proof of this lemma is omitted but can be found in [START_REF] Daoudi | Multiscale autoregressive models and wavelets: Extended version[END_REF].

Lemma 1: Let be an integer in , then

where we have [START_REF] Krim | Multiresolution analysis of a class of nonstationary processes[END_REF], as shown at the bottom of the page, and [START_REF] Lindquist | On the stochastic realization problem[END_REF], also shown at the bottom of the page. We now prove Proposition 1.

Proof: Define the matrices and as shown in (31) at the bottom of this page, and the triangular matrices and as also shown at the bottom of this page, then ( 27) and [START_REF] Irving | A canonical correlations approach to multiscale stochastic realization[END_REF]


Adel Mesbah
email: adel.mesbah@cea.fr

Stephanie Szenknect

Nicolas Clavier

Hantao Lin

Fabien Baron

Daniel Beaufort

Yann Batonneau

Julien Mercadier

Aurélien

Stéphanie Szenknect

Aurélien Eglinger

Marion Turuani

Philippe Goncalves

Flavien Choulet

Virginie Chapon

Anne-Magali Seydoux-Guillaume

Maurice Pagel

Nicolas Dacheux

Direct synthesis of pure brannerite UTi2O6

Keywords: brannerite, uranium titanate, uranium hydroxide

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In uranium deposits under reducing environments, uraninite (UO2) and coffinite (USiO4) are the main uranium (IV) ore minerals. Brannerite (UTi2O6) represents a significant third ressource of tetravalent uranium in many deposits. However, it is strongly refractory to dissolution in current milling processes and requires a heating step in the chemical processes which significantly increases the cost of uranium extraction from the ore. Brannerite is presently reported to occur in several geological environments such as Elliot Lake (Ontario, Canada) [START_REF] Larocque | current leaching and product recovrey practice at denison-mines-limited[END_REF][START_REF] Hester | Current developments at RIO-ALGOM, ELLIOT-LAKE[END_REF], Mount Isa (Australie) [START_REF] Wilde | Geology and Mineralogy of Uranium Deposits from Mount Isa, Australia: Implications for Albitite Uranium Deposit Models[END_REF][START_REF] Pownceby | Geometallurgy of Australian uranium deposits[END_REF], Kirovograd (Ukraine), Crocker Well (Australia) [START_REF] Gilligan | The extraction of uranium from brannerite -A literature review[END_REF], Domes Region (Zambia) [START_REF] Eglinger | Uranium mobilization by fluids associated with Ca-Na metasomatism: A P-T-t record of fluid-rock interactions during Pan-African metamorphism (Western Zambian Copperbelt)[END_REF] and also in some uraniferous deposits in the Witwatersrand area (South Africa) [START_REF] Lottering | Mineralogy and uranium leaching response of low grade South African ores[END_REF]. Moreover, brannerite is also detected in exploited uranium mines in Australia such as Ranger or Olympic Dam [START_REF] Pownceby | Geometallurgy of Australian uranium deposits[END_REF]. In nature, brannerite is usually found as an amorphous form subsequently to self-radiation damage (metamictization) [START_REF] Charalambous | Chemical and microstructural characterisation studies on natural and heat treated brannerite samples[END_REF][START_REF] Zhang | Recrystallisation of amorphous natural brannerite through annealing: The effect of radiation damage on the chemical durability of brannerite[END_REF]. The annealing of the crystal network requires a thermal treatment at 1000 °C. Additionally, brannerite is also considered as a minor phase entering the composition of several Synroc families, which have been developed in the field of immobilization of actinides from nuclear wastes [START_REF] Yudintsev | Brannerite, UTi2O6: Crystal chemistry, synthesis, properties, and use for actinide waste immobilization[END_REF].

Brannerite is reported to be strongly refractory to dissolution [START_REF] Gilligan | The extraction of uranium from brannerite -A literature review[END_REF]. The uranium recovery often requires the use of aggressive conditions such as 0.5 mol L -1 H2SO4 above 75 °C. One explanation of this property arises from the presence of titanium in the structure, which can lead to the formation of titanium hydroxide then titanium oxide acting as a passivation layer on the surface of the mineral when saturation conditions are reached in the dissolution media.

Therefore, different dissolution studies developed on synthetic and natural brannerite suggested a strong benefit coming from the use of oxidative conditions associated to the presence of Fe 3+ species.

From a structural point of view, brannerite crystallizes in a monoclinic system with the C2/m space group. The structure consists of infinite layers formed by TiO6 octahedra separated by UO6 octahedra down the c axis as viewed in Figure 1. As instance, it was obtained by firing a mixture of uranium and titanium dioxides at high temperature. Indeed, Helean et al. [START_REF] Helean | Enthalpies of formation of U-, Th-, Ce-brannerite: implications for plutonium immobilization[END_REF] obtained brannerite samples by heating the mixtures (UO2+TiO2) at 1200°C for 300 h under CO -5% / CO2 -95% atmosphere. Other chemical routes involving the precipitation of low-temperature precursors were also developed. First, we can cite the case of the "Alkoxide/nitrate" route [START_REF] Ringwood | Radioactive Forms for the Future[END_REF][START_REF] Vance | Crystal Chemistry and Stabilization in Air of Brannerite, UTi2O6[END_REF] and the "Acetate" route developed by Hussein et al [START_REF] Hussein | Synthesis and dissolution studies of brannerite, a uranium containing mineral[END_REF] then used by Charalambous et al. [START_REF] Charalambous | An investigation on the dissolution of synthetic brannerite (UTi2O6)[END_REF]. All these chemistry routes require extended mechanical grinding step during the fabrication process, which are usually considered as a heavy step when handling radioactive materials. In addition, this grinding process cannot guaranty the homogenization in terms of cationic distribution (U, Ti), which could result in the formation of UO2 and/or TiO2 as secondary phases. Moreover, the latter techniques use U(VI) based reactants as starting precursors. Additionally, the working atmosphere under which the thermal treatment is performed was not always specified in the literature, except by Vance et al. [START_REF] Vance | Crystal Chemistry and Stabilization in Air of Brannerite, UTi2O6[END_REF] who mentioned the need to perform the experiments under low oxygen content (pO2 < 10 -5 atm). Consequently, the oxidation state of uranium in the synthesized brannerite is not always well established. Collela et al. [START_REF] Colella | Determination of the uranium valence state in the brannerite structure using EELS, XPS, and EDX[END_REF] performed an exhaustive study on the valence state of uranium in natural and synthetic brannerite samples using a combination of techniques. They showed that even in synthetic samples prepared by the alkoxide-nitrate processing route, the average valence state of uranium was 4.3 due to the presence of a significant amounts of U 5+ and U 6+ in the structure and reached 4.4 to 4.7 in natural samples.

Considering that the presence of secondary phases (mainly UO2) resulting from the method of synthesis and the oxidation state of uranium in brannerite have a potential role in the alteration mechanisms and in the release rates upon exposure to aqueous fluids, there is a need to develop new ways of syntheses avoiding the precipitation of actinides as less refractory secondary phases and allowing the control of the oxidation state of uranium. Ultimately, such a synthetic sample could be used as a reference material either to study the mechanisms of dissolution of ores containing brannerite, to identify brannerite in ores using spectroscopic techniques or to determine the valence states of uranium in natural and substituted brannerites.

With the aim, this paper is devoted to the preparation and characterization of pure brannerite samples following three synthesis protocols reported in the literature namely alkoxide/nitrate method (1), acetate method (2) and dry chemistry route from UO2 + TiO2 mixtures [START_REF] Wilde | Geology and Mineralogy of Uranium Deposits from Mount Isa, Australia: Implications for Albitite Uranium Deposit Models[END_REF]. Several drawbacks associated to the three chemistry routes were then solved by the development of a new method based on the precipitation of a uranium (IV) and titanium (IV) hydroxide mixture [START_REF] Pownceby | Geometallurgy of Australian uranium deposits[END_REF].

Experimental section

Synthesis.

The reactants during the synthesis, i.e. NH4OH (≥ 99%), titanium alkoxyde (≥ 97%), TiO(SO4).nH2O (29% TiO2 basis), and H2C2O4.2H2O (≥ 99%) were of analytical grade and were purchased from Sigma Aldrich, uranyl acetate was home made. The uranium source was kindly provided by CETAMA France. Uranium (IV) chloride was obtained by dissolving uranium metal chips in 6 mol L -1 HCl. For U(VI) supply, a concentrated uranyl nitrate solution was used. The final concentration of each solution was determined by ICP-AES. In order to synthesize brannerite, the four different chemistry routes listed below were considered.

1.

"Acetate route" [START_REF] Hussein | Synthesis and dissolution studies of brannerite, a uranium containing mineral[END_REF][START_REF] Charalambous | An investigation on the dissolution of synthetic brannerite (UTi2O6)[END_REF]: this method consisted in simultaneous dissolution of uranyl acetate (2.33 mmol) and titanyl sulphate dihydrate (4.66 mmol) in 1 L of a solution containing 120 g oxalic acid . The mixture was dried in air at 200 °C then heated at 600 °C under Ar/H2 for 5 h in order to eliminate the organic part and other residues and with the goal to obtain oxides (UO2 and TiO2). The obtained powder was ground, pressed into pellets then thermally heated at 1300 °C for 96 h.

2.

"Alkoxide/nitrate" route [START_REF] Ringwood | Radioactive Forms for the Future[END_REF][START_REF] Vance | Crystal Chemistry and Stabilization in Air of Brannerite, UTi2O6[END_REF]: this method consisted in the mixing of solutions containing uranyl nitrate (1.16 mmol) and titanium alkoxide (2.32 mmol) in 100 mL of water. The solution was dried in air at 200 °C, heated in air at 750 °C for 1 h leading to the elimination of nitrates and residues coming from the organic part leading also to the formation of oxides (U3O8 + TiO2). The obtained powder was then wet milled for 16 h, pressed into pellets then fired at 1300 °C for 14 h.

3.

"Dry chemistry route". For this method, a mixture containing powdered UO2 and TiO2 was mechanically ground during 16 h at room temperature, pressed into pellets then thermally heated at 1300 °C for 96 h.

4.

"Hydroxide precipitation route". The hydroxide method was adapted from the synthesis protocol developed by Martinez et al [START_REF] Martinez | An original precipitation route toward the preparation and the sintering of highly reactive uranium cerium dioxide powders[END_REF], which was initially used for the synthesis of uranium dioxide. In this frame, three experiments were carried out.

The first one consisted in mixing solutions of U (IV) chloride (2.33 mmol L -1 ) and of Ti-alkoxyde solution (4.66 mmol L -1 ) with 17 mL of deionized water. Then, an excess of NH4OH (46.6 mmol) was added leading to the formation of a uraniumtitanium bearing hydroxide brownish gel. After 30 min of stirring, the mixture was washed twice by centrifugation in water and once in ethanol. For the second test, a Ti excess of 3 mol.% of titanium was added in the mixture. The third set consisted in the addition of titanium alkoxyde in the solution after the precipitation of U(OH)4.nH2O.

For all the conditions, the final powder dispersed in ethanol was dried under vacuum at 40 °C leading to de-agglomerated powder having a large specific surface area (larger than 100 m 2 .g -1 ) [START_REF] Martinez | An original precipitation route toward the preparation and the sintering of highly reactive uranium cerium dioxide powders[END_REF]. The powder was then compacted into a green pellet then placed in a furnace at 1300 °C for 96 h under argon atmosphere.

Whatever the method followed in this work, the thermal treatment at 1300°C under Ar atmosphere was conducted on pellets in order to favor the diffusion between the elements with the goal to obtain pure brannerite.

Powder X-Ray Diffraction (PXRD)

The obtained samples were firstly ground using an agate mortar, then the resulting powders were analyzed by PXRD using Bruker D8 Advance diffractometer equipped with copper radiation (Cu Kα1,2, λ = 1.54184 Å) and using the reflection geometry. The powders were placed in special confined sample holders in order to avoid any radioactive contamination.

The PXRD patterns were collected in the angular range of 5 ≤ 2 θ ≤ 100° with a total counting time of about 3 h. In addition, PXRD pattern of pure silicon standard was collected in similar conditions in order to extract the instrumental function. The collected data were refined by the Rietveld method with the use of the Fullprof_Suite package [START_REF] Frontera | FullProf as a new tool for flipping ratio analysis[END_REF]. During the refinement, different profile and structure parameters were allowed to vary. Moreover, an anisotropic size model was added for each phase to simulate the microstructural effect.

NIR

The visible and near-infrared spectrum of the synthetic brannerite was acquired in the 350-2500 nm range using the field ASD spectrometer TerraSpec ® . The spectrometer was equipped by one Si detector (350-1000 nm range) and two InGaAs detectors (1000-2500 nm) having a spectral resolution of 4 nm and 8 nm, respectively. The acquisition was conducted using a contact probe including a white light source (halogen bulb). The presented spectrum of the synthetic brannerite resulted from an average of 35 scans.

Raman spectroscopy

Raman spectra were recorded by the means of a Horiba -Jobin Yvon Aramis apparatus equipped with an edge filter and using a Nd:YAG laser (532 nm). In order to avoid any laserinduced degradation of the compound, the power was turned down to about 5 mW by the means of an optical filter. The laser beam was then focused on a small fraction of powder simply deposited on a glass lamella using an Olympus BX 41 microscope. A ×50 objective with a numerical aperture of 0.8, resulting in a spot size of about 1 µm 2 was used. The scattered Raman light was collected in a 180° backscattering geometry and dispersed by a grating of 1800 grooves/mm after having passed a 150 µm entrance slit, resulting in a spectral resolution lower than 1 cm -1 . For each spectrum, a dwell time of 15 s was considered with an average of 4 scans.

Before analysis, the apparatus was calibrated with a silicon wafer, using the first-order Si line at 520.7 cm -1 with an accuracy lower than 1 cm -1 .

The band component analysis was carried out by the means of the Jandel Peakfit software, using pseudo-Voigt functions (Gaussian-Lorentzian ratio systematically higher than 0.7) with the minimum number of components. Correlation coefficient R 2 greater than 0.994 was obtained.

Results and discussion

Whatever the protocol considered in routes (1), ( 2), (3), brannerite (UTi2O6) was always obtained as the major phase. PXRD patterns corresponding to the powders obtained following the three first protocols are reported in Figure 2. However, the prepared samples were polyphase since the presence of uranium dioxide (UO2) was always observed as secondary phase. No sign of unreacted titanium (e.g. present as TiO2) was evidenced. For each sample, the weight contents of each phase (brannerite and UO2) and the unit cell parameters were determined by Rietveld refinement (Table 1). The uranium dioxide content was clearly dependent on the synthesis route; varying from 1 wt.% (dry chemistry route) up to 25 wt.% (acetate route). The results obtained from the Rietveld refinement confirmed the formation of brannerite (as the main phase) and UO2 in the prepared powder. The methods based on the precipitation of a low temperature precursor (i.e. acetate and alkoxide/nitrate routes) showed the presence of large amounts of UO2 compared to that obtained by dry chemistry route (i.e. mixture of UO2 and TiO2). This is certainly due to some difficulties in the weighing of the appropriate amounts of titanium (either under the form of TiOSO4.nH2O and/or Ti[OCH(CH3)2]4) caused by variable hydration ratios and hygroscopic character of the samples that induce non-stoichiometric mixtures of U and Ti with a lack of Ti. Moreover, the successive steps of evaporation and calcination could influence the homogeneity of the starting mixtures of precursors. Although dry-chemistry route allows precise weighing of the starting powders, it did not lead to a singlephase material maybe due to lack of reactivity of the oxides powders and slow diffusion of both titanium and uranium in the starting mixtures.

In addition, the refined unit cell parameters were compared to those reported in the literature. In the case of the application of the "acetate method" (1), the unit cell volume of the brannerite was found to be 224.02(1) Å 3 , which is close to the value reported by J. Szymanski.

et al, [START_REF] Szymanski | A crystal-structure refinement of synthetic brannerite, UTi2O6, and its bearing on rate of alkaline-carbonate leaching of brannerite in ore[END_REF] (i.e. 224.14 Å 3 ). On the contrary, the unit cell volumes obtained for "alkoxide/nitrate" and "dry chemistry" routes were found to be slightly lower. This could be explained by the presence of small amounts of U(VI) within the structure. In the case of the "acetate route", the calcination of the precursor was performed under Ar/H2, which led to the reduction of uranium into its tetravalent form. In the case of the "alkoxide/nitrate" and "dry chemistry" routes, uranium was either introduced in higher oxidation state than 4+ because of the milling step leading to the formation of U3O8. Then after, U3O8 decomposed under low oxygen content into UO2+x above 1000 °C. Therefore, regardless the followed method, the obtained brannerite was not pure and contained various amounts of UO2 as by-product. Moreover, there are some practical drawbacks associated with these protocols such as the evaporation of the initial solution at 200 °C, a milling step for more than 12 h involving a radioactive element, which could increase the risk of contamination.

Since the samples prepared by the three first methods did not result to a single phase, a new method based on the precipitation of uranium and titanium hydroxide precursor followed by its direct heating at 1300 °C was developed without any prior heating or grinding step. The PXRD patterns obtained for the three types of "hydroxide precipitation" methods are gathered in Figure 3. The analysis of the PXRD patterns reported in Figure 3 showed that whatever the "hydroxide precipitation" route considered, brannerite (UTi2O6) was always obtained as the main phase. However, when using the first or the third variety, the formation of impurities (i.e.

UO2 and TiO2) was detected. The weight contents of secondary phases as well as the unit cell parameters refined by the Rietveld method are reported in Table 2.

Table 2.

Refined unit cell parameters for the synthesized UTi2O6 and weight amount (wt %) found for each phase. The refined unit cell parameters of the three brannerite samples prepared using the hydroxide routes are in good agreement with those reported in the literature (224.14 Å 3 ) [START_REF] Szymanski | A crystal-structure refinement of synthetic brannerite, UTi2O6, and its bearing on rate of alkaline-carbonate leaching of brannerite in ore[END_REF].

a (Å) b (Å) c (Å) β (°) V (Å
In fact the use of U(IV) based reactants as starting precursors certainly prevents the oxidation of uranium in the brannerite structure. Moreover, from the synthesis (4.1) the weighing of the stoichiometric amounts led to the formation of UO2 considered as an impurity. This could be explained by the underestimation of the introduced titanium amount into the reaction because of the hydrolysis and polycondensation of Ti, or by a possible insertion of titanium into the uranium site. Therefore, in order to avoid any lack of titanium in the starting mixture, a slight excess of titanium (i.e. 3 mol.%) was added during the preparation of the precursor. From the results reported in Figure 3 and in Table 2, the synthesis led to pure brannerite with no need of additional grinding and/or re-heating step. On the contrary, when using the third "hydroxide precipitation" variety, the uranium dioxide content in the final sample was significantly higher than that obtained for the two other varieties. This could result from a degradation of the homogeneity of the mixture of the precursors due to the separated precipitation of uranium and titanium. In this latter case, the benefit coming from the use of the "hydroxide precipitation" route was clearly counterbalanced. This benefit relies on the preparation of a homogeneous and intimate mixture of nanometric hydroxide precursors. The large surface area developed between Ti and U(IV) hydroxide particles associated to the nanometric size of Ti and U(IV)-rich domains give the mixture its high reactivity.

Near-Infrared spectroscopy

The visible and near-infrared spectrum of the synthetic brannerite includes several weak absorption bands in the 350-1000 nm wavelength range (Figure 4). This result is in contradiction with the studies of Finnie et al. [START_REF] Finnie | Examination of U valence states in the brannerite structure by near-infrared diffuse reflectance and X-ray photoelectron spectroscopies[END_REF] and Vance et al. [START_REF] Vance | Crystal Chemistry and Stabilization in Air of Brannerite, UTi2O6[END_REF], where the authors did not observe any spectral features within this wavelength range for synthetic brannerite, which was prepared by alkoxide/nitrate route. However, the position and the relative intensity of bands

Raman spectroscopy

The Raman spectra collected for the various synthetic samples of UTi2O6 were all found to present similar features. An example is provided in Figure 5. As the structure of brannerite is composed by UO6 and TiO6 polyhedra, the modes that can be observed mainly correspond to the vibrations of oxygen atoms. Indeed, the factor group analysis provided by Zhang et al. [START_REF] Zhang | Raman spectroscopic study of natural and synthetic brannerite[END_REF] leads to:

Γ = 8Ag + 4Bg + 5Au + 10Bu (3)
Among these modes, 2Ag + Bg + 2Au + 4Bu accounts for external translation motions, while the remaining 6Ag + 3Bg, and 3Au 6Bu, are Raman and infrared active, respectively.

All the bands observed in the 100-1000 cm -1 domain were found to be in good agreement with those already reported for synthetic samples of brannerite [START_REF] Zhang | Raman spectroscopic study of natural and synthetic brannerite[END_REF], as well as for Ce- [START_REF] Kong | Novel Chemical Synthesis and Characterization of CeTi2O6 Brannerite[END_REF] and Thbearing [START_REF] Zhang | Theoretical and experimental Raman spectroscopic studies of synthetic thorutite (ThTi2O6)[END_REF] counterparts. Particularly, the stretching vibrations of the (Ti-O-Ti) moiety lead to the most intense band at 760 cm -1 for the antisymmetric mode, and to a weak band at 488 cm -1 for s(Ti-O-Ti). Based on an analogy with isostructural vanadates [START_REF] Baran | Raman-Spectra of Some M-Li-V2O6 Brannerite-Type Metavanadates[END_REF], the two bands located at 571 and 632 cm -1 were then assigned to the Ti2O2 bridges corresponding to the edges shared between two TiO6 octahedra. The assignment of the three bands observed at 272, 331 and 374 cm -1 appeared to be more complicated but was sometimes described as the result of the combination of (Ti-O-Ti) bending with stretching modes of UO6 polyhedra [START_REF] Kong | Novel Chemical Synthesis and Characterization of CeTi2O6 Brannerite[END_REF]. Finally, the vibration modes occurring below 200 cm -1 are correlated to lattice external modes.

Compared to natural samples, the most obvious difference is the absence of vibration bands correlated with the stretching vibrations of the uranyl entity (UO2 2+ ) [START_REF] Frost | Raman spectroscopic study of the uranyl titanate mineral brannerite (U,Ca,Y,Ce)(2)(Ti,Fe)(2)O-6:effect of metamictisation[END_REF], which confirms that uranium is fully incorporated in the tetravalent oxidation state in the synthetic brannerite.

Also, one must note that no band associated with the TiO2 by-product (such as that pointed out by Zhang et al. around 435 cm -1 [START_REF] Zhang | Raman spectroscopic study of natural and synthetic brannerite[END_REF]) was observed. On this basis, the Raman spectroscopy argues for the formation of a pure and single-phase UTi2O6 sample after firing at 1300 °C of the precursor obtained by "hydroxide precipitation" route (4.2).

Conclusion

A new direct route was developed to synthesize pure brannerite. This protocol consisted of the precipitation of powdered hydroxide (An,Ti)(OH)4.nH2O precursor which was submitted to compaction into pellets and firing at 1300 °C for three days in the presence of low oxygen content (Argon atmosphere). The precipitation of the hydroxide powder allowed the preparation of highly reactive powder having a large specific surface area (> 100 m².g -1 ) which facilitated the homogenization of the powder. This allowed the elimination of the heavy grinding step of radioactive materials. Also, the application of the hydroxide method conducted to pure powder conversely the previous protocols reported in the literature. In addition, the protocols cited in the literature involve the use of hexavalent uranium as starting reactant or of U3O8 after grinding of UO2 in the mixture of oxides. This could induce the presence of various amounts of U(VI) within the brannerite samples, conversely to the hydroxide method which directly involved U(IV) as a reactant and thus should leave to the formation of U IV Ti2O6. Finally, spectroscopic measurements on pure brannerite confirmed the absence of impurities such as TiO2 and UO2 in the prepared samples and confirmed the presence of tetravalent uranium only.

Figure 1 .

 1 Figure 1. Representation of the crystal structure of UTi2O6 down the b axis

Figure 2 .

 2 Figure 2. PXRD patterns obtained for brannerite samples prepared by "acetate route" (1), "alkoxide/nitrate route" (2) and dry chemistry route (3)

Figure 3 .

 3 Figure 3. PXRD patterns collected for the samples prepared by the three varieties of "hydroxide precipitation" route (4). U(IV) + Ti-alkoxyde (4.1), with addition of 3 mol.% excess of titanium (4.2), with addition of Ti-alkoxyde after basification (4.3).

  observed in the present spectrum of the synthetic brannerite prepared by the new hydroxide route (4.2) match very well with the spectra of U 4+ -doped thorutite (Th1-xUxTi2O6) presented by Finnie et al.[START_REF] Finnie | Examination of U valence states in the brannerite structure by near-infrared diffuse reflectance and X-ray photoelectron spectroscopies[END_REF]. The observed absorption bands correspond to the electronic transitions of U 4+ ions in the brannerite structure. No evidence of U 5+ , characterized an intense and sharp absorption band around 1450 nm as described by Vance et al.[START_REF] Vance | Crystal Chemistry and Stabilization in Air of Brannerite, UTi2O6[END_REF], Finnie et al.[START_REF] Finnie | Examination of U valence states in the brannerite structure by near-infrared diffuse reflectance and X-ray photoelectron spectroscopies[END_REF] and Zhang et al.[START_REF] Zhang | Development of brannerite glass-ceramics for the immobilization of actinide-rich radioactive wastes[END_REF] was observed in the spectrum of synthetic brannerite.

Figure 4 .

 4 Figure 4. Visible / Near-IR spectrum of the synthetic brannerite (UTi2O6) prepared by the hydroxide route (4.2). The position of the spectral features are expressed in nm.

Figure 5 .

 5 Figure 5. Raman spectrum of single-phase synthetic UTi2O6 brannerite obtained by "hydroxide precipitation" (4.2), i.e. in the presence of 3 mol.% excess of titanium in the starting mixture.

Table 1 .

 1 Refined unit cell parameters and weight contents of UTi2O6 and UO2 (expressed in wt.%). Obtained for the samples prepared by "acetate route" (1), "alkoxide/nitrate route" (2) and dry chemistry route (3).

		a (Å)	b (Å)	c (Å)	β (°)	V (Å 3 )	VUO2 (Å 3 )	wt.% UTi2O6	wt.% UO2
	# 1	9.8144(3)	3.7672(2)	6.9193(3)	118.87(1)	224.02(1)	162.56(1)	75 ± 2	25 ± 2
	# 2	9.8176(2)	3.7606(1)	6.9050(1)	118.90(1)	223.17(1)	161.55(1)	87 ± 1	13 ± 1
	# 3	9.8171(2)	3.7591(1)	6.9017(2)	118.94(1)	222.88(1)	161.79(4)	99 ± 1	1 ± 1
	[19]	9.8123(15)	3.7697(6)	6.9253(9)	118.957(6) 224.14

Acknlowdgements:

The present work was supported by the UTiLE project funded by NEEDS Ressources program (CNRS/CEA/AREVA)