André Ducrot

Yann Dumortier

Isabelle Herlin

Vincent Ducrot

Real-Time Quasi Dense Two-Frames Depth Map for Autonomous Guided Vehicles

This paper presents a real-time and dense structure from motion approach, based on an efficient planar parallax motion decomposition, and also proposes several optimizations to improve the optical flow firstly computed. Later, it is estimated using our own GPU implementation of the well-known pyramidal algorithm of Lucas and Kanade. Then, each pair of points previously matched is evaluated according to the spatial continuity constraint provided by the Tensor Voting framework applied in the 4-D joint space of image coordinates and motions. Thus, assuming the ground locally planar, the homography corresponding to its image motion is robustly and quickly estimated using RANSAC on designated well-matched pairwise by the prior Tensor Voting process. Depth map is finally computed from the parallax motion decomposition. The initialization of successive runs is also addressed, providing noticeable enhancement, as well as the hardware integration using the CUDA technology.

I. INTRODUCTION

This contribution addresses the scene perception for intelligent mobile vehicle systems using a monocular video sensor. Perception issues for Autonomous Guided Vehicle consist in estimating the free driving space as well as detecting moving obstacles. Depending on sensors and architecture, such systems can have different price, accuracy and speed characteristics. On one hand, there are active range-finders sensors. Among them, lasers provide high spatial resolutions data at high scanning speeds for a quite expensive price, whereas other cheaper sensors like sonars, are limited to parking applications owing to their range. In addition, the presence of several active sensors in the same area may disrupt measure acquisition. On the other hand, video sensors appear well adapted for automotive applications, due to the rich 2-D information contained in a single image and the 3-D information inferred by two. To date, research has mostly focused on the perception from fix and mobile cameras in static or constrained dynamic scenes [START_REF] Talukder | Real-time detection of moving objects in a dynamic scene from moving robots vehicles[END_REF], [START_REF] Irani | A unified approach to moving object detection in 2d and 3d scenes[END_REF]. Recovering the 3-D from a monocular sensor is considered as an highly computational task. Most existing approaches do not allow real-time constraints [START_REF] Zhang | Consistent Depth Maps Recovery from a Video Sequence[END_REF] while the others correspond to sparse feature-based methods for Simultaneously Localization And Mapping (SLAM) [START_REF] Eade | Edge landmarks in monocular SLAM[END_REF]. In the case of mobile robotic applications, the minimal sufficient information is described by the Extended Minimal Model [START_REF] Dumortier | 4-D Tensor Voting Motion Segmentation for Obstacle Detection in Autonomous Guided Vehicle[END_REF] which includes the road surface estimation and the mobile obstacles tracking. Most featurebased approaches first compute a road displacement model between two frames, and then perform a dense comparison between the second image and the first one, warped using the transformation previously computed to compensate the camera motion. Some important issues inferred from such methods are discussed in section III-A. Recent technological advances, especially regarding the High Performance Computing on common integrated GPU, using CUDA, CTM or OpenCL API, or on dedicated architecture such as FPGA and DSP, allow performing dense motion fields with real-time constraints [START_REF] Marzat | Real-Time Dense and Accurate Parallel Optical Flow using CUDA[END_REF], [START_REF] Wedel | Structure and Motionadaptive Regularization for High Accuracy Optic Flow[END_REF]. Thus Structure From Motion approaches can now be considered to recover the overall depth of an observed scene.

The rest of the document is organized as follows. The section II presents optical flow computation and its real-time integration on GPU, as well as an efficient method to filter the final motion field using the Tensor Voting framework. The section III describes how to identify the ground plane and extract the translational component of the camera's displacement from the image motion. It also details the way to compute a quasi dense depth map from this motion field, in a real-time manner. Then in section IV, we propose two different optimizations to improve the robustness of the depth map estimated in our approach. At last, conclusions and directions for future works are given in section V.

II. ROBUST REAL-TIME OPTICAL FLOW

So far, many efforts in computer vision have been focused on retrieving the 3-D location of 2-D image points. Depending of the considered application and its relative constraints, the "inverse projection" of 2-D points in the Euclidean space can be classified in different processes such as:

• Matching sparse points or structures of interest according to some specific descriptors, trackers and spatial constraints.

• Directly estimating the depth map from the epipolar geometry, given the baseline between two or more cameras operating simultaneously, and using multi-views constraints.

• Approximating the image motion field, by using a temporal photometric constraint, that consists in computing the optical flow.

The latter commonly used the brightness constancy constraint, also called optical flow constraint, which assumes that the pixel intensity of an image point does not change with time.

It can be written in a discrete manner as follows:

I 1 (x 1 ) = I 2 (x 2 ) = I 2 (x 1 + △x 21 ) , (1)
with x i the image coordinates of a point X and I j (x i ) the brightness of its pixel at time t i and t j , respectively. △x ji describes the discrete image motion of X between t i and t j .

In the continuous case, with dI/dt = 0, the differential form of ( 1) is given by:

dI dt = dI (x (t) , y (t) , t) dt = I x dx dt + I y dy dt + I t = I x I y dx/dt dy/dt + I t = ∇I T ω + I t (2)
where I x , I y and I t represent the partial spatial and temporal derivatives describing the image gradient ∇I, and ω the image motion.

A. Real-time Lucas and Kanade tracker 1) Lucas and Kanade: Given any image point x, the Eq. ( 2) only provides an estimation of its 2-D motion along spatial gradient direction. This issue, well-known as the aperture problem, can partially be solved by integrating the optical flow constraint on the neighborhood Ω ROI of x. In [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF] Lucas and Kanade assume the motion uniform over Ω ROI to minimize the following expression:

J (ω) = Ω ROI (∇I ω + I t ) 2 , (3)
and write from (3) the over-determined system below:

     I x1 I y1 I x2 I y2 . . . . . . I xn I yn      A • ω x ω y ω = -      I t1 I t2 . . . I tn      b , (4)
that can be easily solved using the least square method, such as:

ω = A T A -1 A T (-b) .
The solution is optimal for all pixels in Ω ROI and thus quite sensitive to any assumption violation:

• if Ω ROI includes pixels which do not have the same motion than x, • or if the considered image motion is too large regarding

|Ω ROI |.
Such issues can be fixed, first by a prior investigation on the optimal size of |Ω ROI |, using synthetic data describing the same kind of motion than into the considered application (Fig. 1), and by applying a pyramidal coarse-to-fine scheme as described in [START_REF] Bouguet | Pyramidal Implementation of the Lucas Kanade Feature Tracker: Description of the Algorithm[END_REF]. 2) CUDA implementation: In case of vehicles operating in an open-environment, the perception process must be performed "on-the-fly", respecting as well a real-time constraint. However, current implementations of the pyramidal Lucas Kanade tracker do not perform in less than few seconds -depending of the CPU used -to provide a dense motion field on grayscale mid-resolution images (640 × 480 pixels). We proposed in [START_REF] Marzat | Real-Time Dense and Accurate Parallel Optical Flow using CUDA[END_REF] a real-time solution, using Nvidia CUDA technology, that executes the Lucas Kanade algorithm on a common GPU. This integration takes advantage of multi-processor architecture since optical flow can be computed independently for each pixel.

A comparison of the results obtained with our C++/CUDA library and the OpenCV version is presented Fig. 2, where the upper-left image corresponds to the first frame of the input video sequence. As showed in the upper-right image, the standard optical flow representation, that consists in a uniform distribution of 2-D vectors, does not allow visualizing efficiently the motion field smoothness. So we densly render the optical flow in the following illustrations with a color map coding the angle and the norm of each motion vector with respectively the hue and the brigthness pixel values.

By comparing with the OpenCV result (Fig. 2, second row), our implementation provides some interesting improvements, mainly due to the bilinear sampling of the motion field between two levels of the pyramid, instead of the nearestneighbor interpolation used by OpenCV to reduce the computational time of its solution. Indeed, modern GPUs provide texture units able to perform this kind of bilinear interpolation without additional cost. The optical flow is computed 100 times faster with our method than on dual-core CPU, that is about 30 milliseconds.

B. Motion saliency and Tensor Voting

In some degenerated cases, such as textureless areas or occluded regions, the system (4) is not able to provide a good estimation of the image motion, and we need to compute a confident measure of the optical flow. However, since the least squares solution is optimal for each pixel into Ω ROI , the optical flow constraint is supposed to be locally observed and thus, no photometric measurement can afford to evaluate the motion field previously computed. Nonetheless, by assuming that disconuities only occur at 3-D object boundaries, we can assume the motion field globally smooth and so apply a spatial continuity constraint to check any wrong estimation.

In this way, the following section describes use of 4-D Tensor Voting framework, initially presented by G. Medioni and M. Nicolescu [START_REF] Nicolescu | Motion Segmentation with Accurate Boundaries -A Tensor Voting Approach[END_REF], to quantitatively assess the optical flow.

1) 2-D Tensor Voting:

Tensor Voting is a unified framework which has been developed by G. Medioni since the 90's. This formalism, based on tensor calculus for representing data and on a nonlinear voting process for their communication, identifies the geometrical structures described by the layout of a sparse and potentially noisy N-D dataset. It also provides for each point a saliency measure on its belonging to the structure inferred. As the N-D case is a generalization of the initial 2-D framework, we begin with this case before detailing the 4-D process.

In a 2-D space, Tensor Voting attempts to retrieve, from a set of sparse and isotropic data, geometrical structures such as curve elements and points. To do this, it uses the neighborhood layout of each considered input site to constrain their identification. Second order representation, in the form of a second order, symmetric, non-negative definite tensor, allows to encode the structural information of the input data, and to propagate this information to its neighborhood according to a simple tensor accumulation process.

Every 2-D tensor T can be represented using its quadratic form T, that is a symmetric 2x2 matrix which also graphically defines an ellipse. According to the matrix algebra, T can be decomposed in the following way:

T = (λ 1 -λ 2 ) ê1 êT 1 stick + λ 2 ê1 êT 1 + ê2 êT 2 ball . ( 5
)
where λ i are the eigen values in decreasing order (λ i ≤ λ i+1 ) and êi the corresponding unit eigen vectors. Each of the two terms corresponds to a specific elementary 2-D tensors: the first describes a degenerate elongated tensor and the second an isotropic one, named respectively stick and ball tensors. In 2-D Tensor Voting framework, data are encoded according to their isotropy with such unitary tensors. Thus, a curve element is coded by a stick tensor belonging to the normal of the considered curve while a point (without any structural information) is represented by a ball tensor.

The communication step is an accumulation process that involves a voter T p and a votee (receiver) T i . The voter infers on the votee according to the distance l between them. We voluntary limit the tensor voting scheme to its isotropic version, the ball vote, while the complete framework also takes into account the relative orientation between the voter and the votee. Each input site, originally coded by a unit ball tensor, communicates its relative position to its neighborhood using stick tensors that defines the normal direction to the straight line joining the votees. The stick is weighted with a Gaussian decay function that models its decreasing effect on its neighborhood with the distance l. Let T i,t be the tensor T i at time t and V i (T j,t ) the vote it receives from a ball tensor T j,t of its neighborhood Ω. For each tensor T the voting process can be written as follows:

T p,2 = T p,1 + Ω i V p (T i,1 ) ,
where the quadratic form of V , that also corresponds to a tensor, is given by:

V = e - 0 @ v σ 1 A 2 I -v v T , (6)
with v the vector joining the voter and the votee, and σ, a constant which defines the scope of the vote in the decay function.

Extracting the structural information from the resulting tensors is done by decomposing them according to [START_REF] Talukder | Real-time detection of moving objects in a dynamic scene from moving robots vehicles[END_REF]. The saliency measures λ 1 -λ 2 and λ 2 provide the preferred orientation of each tensor and so, the geometrical element it belongs to. As the method is non-iterative, it can be processed in a constant time complexity by parallel implementation.

2) Saliency measure in the joint-space (x, y, vx, vy): As introduced above, we suppose that the 3-D motion of any objects is continuous, as well as their motion in the focal plane, that just depends of the camera frame rate. Thus, the camera displacement provides a spatial collection of continuous motion areas in the focal plane that actually describe smooth surfaces in the 4-D joint-space (x, y, v x , v y ) of image locations and motions. Fig. 3 illustrated such area in the 3-D sub-space (x, y, v x ). Using the Tensor-Voting-framework in 4-D, as described in [START_REF] Nicolescu | Motion Segmentation with Accurate Boundaries -A Tensor Voting Approach[END_REF], we propose to focus our attention on the discontinuities present in this space, that translate both area boundaries and wrong motion estimations. The N-dimensional Tensor Voting approach is an extension of the original framework, and the quadratic form of each 4-D tensor can also be decomposed in 4 elementary tensors as in ( 5):

T = (λ 1 -λ 2 ) ê1 êT 1 stick + (λ 2 -λ 3 ) ê1 êT 1 + ê2 êT 2 disk 2D + (λ 3 -λ 4 ) ê1 êT 1 + ê2 êT 2 + ê3 êT 3 disk 3D + λ 4 ê1 êT 1 + ê2 êT 2 + ê3 êT 3 + ê4 êT 4 ball (7)
where 2-D disk corresponds to the elementary tensor coding geometrical structure of type surface.

The first step consists in quantifying the relationship between the saliency measure obtained at the end of the Tensor Voting and the presence of noise into the input motion field previously computed. The graphic Fig. 4 displays the average norm and angular optical flow error relative to the normalized saliency value λ 2 -λ 3 computed on the synthetic video sequence Yosemite, with the optimal parameters found section II-A1. We can notice a threshold appears clearly around the value λ 2 -λ 3 = 0.5, after which it becomes impossible to distinguish motion discontinuities and noise. We use this value to threshold the motion field.

The bottom line Fig. 4 shows resulting motion field after applying this threshold on the surface saliency for our input video sequence. Smooth surfaces that correspond to continuous areas are preserved while almost all the noise is removed.

III. DEPTH MAP

We now assume that optical flow is estimated and refined as presented before, which includes computation of a reliability map corresponding to the saliency measure for each displacement vector. This section addresses the problem of extracting in real-time depth information from 2-D motion captured aboard a mobile architecture. We propose to robustly estimate the ground plane as a first step, before computing the parallax motion in an efficient planar parallax decomposition, and finally inferring the depth map. The ground is supposed locally planar.

A. Robust and real-time ground estimation

To respect the real-time constraint, most of existing ground segmentation methods are feature-based. The general approach extracts road features and match them in two frames to compute the homography H induced by the road plane. Then, it identifies which image parts correspond to the ground, by computing a photometric error. This error is usually a function of the distance between the first image, warped according to the homographic transformation previously estimated, and the second one: Err photo = dist (I 2 (x) , I 1 (Hx)). Such a distance is supposed to be null for all points lying on the ground and non determined otherwise. The result is actually only consistent at gradient discontinuities and very sensitive to homography estimation. The top-left image Fig. 5 illustrates such a photometric error using the l 2 -norm on our video sequence: notice the noise on the road as well as poorly defined ground boundaries.

Since we can estimated the optical flow, we are also able to compute a more reliable error function, corresponding to the warping distance. It simply consists of the distance between two motion fields, the optical flow and the homographic transformation: Err warp = dist (LK (x) , Hx). The top-right image in Fig. 5 shows the l 2 -warping distance map obtained with the same homography matrix as used to compute photometric error (top-left image). The result can be interpreted and allows easily distinguishing the ground limits up to the horizon line. In addition, the measure is robust to thresholding, as illustrated on the graphic Fig. 5 where segmented road area appears stable despite a wide range of threshold values. The homography H induced by the ground displacement between a couple of input frames is robustly computed with RANSAC, which performs a number of iterations that depends on the ratio of outliers in the image correspondences. We propose to use the Tensor Voting saliency map to select the best matches (regarding their motion continuity in the 4-D joint-space previously described) in each cell of a quantization grid, as RANSAC input. In this way, we can reduce the number of outliers corresponding to mismatched points and so, the number of iterations to estimate H.

B. Absolute depth map

Following the general structure from motion scheme, we now infer depth map from the parallax motion. The parallax motion corresponds to the relative displacement of static objects in the focal plane, which is induced by the translational component of the ego-motion (a pure rotation does not provide any parallax). A way to estimate it consists in applying the "plane + parallax" motion decomposition that can be written as follows:

ω = ω π + µ ,
where ω, ω π and µ denote respectively the image motion, the homographic transformation induced by an arbitrary plane of the scene (real or virtual) and the corresponding residual planar parallax motion. The rotational component of the camera displacement is enclosed in the homography and its subtraction from the optical flow provides a part of the translational image motion. In the literature, the homography of an arbitrary plane is used. That results in retrieving only a part of the parallel motion induced by the ego-motion, depending of the plane position. Instead, we compute the rotational homography that allows to estimate the full translational component nmerically more stable for the following steps.

Given the matrix K of camera intrinsic parameters, the homography H induced by a plane located at the distance d from the camera center and defined by its normal n, is expressed by:

H = K R + t 1 d n T K -1 (8)
where R and t are rotation and translation displacements of the camera between two views. Thus, the rotational homography corresponding to the image motion of point located at the infinite distance from the camera center, can be written as:

H ROT = KRK -1 . (9)
The matrix R is here provided by decomposition of the road homography previously estimated (see [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF] for more details).

Let P i = (X, Y, Z)

T and P

′ i = (X ′ , Y ′ , Z ′ ) T denote the
Cartesian coordinates of a scene point with respect to two different camera views, respectively. Let p i = (x, y) T and p ′ i = (x ′ , y ′ ) T respectively denote the corresponding coordinates of the corresponding image points in the two image frames, and t = (t X , t Y , t Z )

T the camera translation between the two views. If we note p wi the image point in the first frame which results from warping the corresponding point p ′ i in the second image by the 2D transformation ω π , when t z = 0:

ω π = (p ′ i -p wi ) . Also, µ =γ t Z d ′ π (e -p wi ) ,
with e the epipole in the first frame, and d ′ π the perpendicular distance from the second camera center to the reference plane. γ = H Z defines the projective structure where H is the perpendicular distance from P i to the reference plane. For more details, see also [START_REF] Irani | A unified approach to moving object detection in 2d and 3d scenes[END_REF]. The relative projective structure of two points P 1 and P i is given by:

γ i γ 1 = µ T i (p wi -p w1 ) ⊥ µ T 1 (p wi -p w1 ) ⊥ , (10)
that corresponds to the relative depth of P i regarding P 1 . It allows computing a relative depth map of the scene. However, since the the distance of camera to the ground is supposed to be constant and measured off-line, the absolute depth can be computed for any image point p i by choosing P 1 lying on the road (but not necessary in the camera field of view). The left image Fig. 6 shows the depth map computed with P 1 chosen such as its projection p 1 would be located at the bottom-right corner of the image. Note the singular line in the direction of the parallax displacement of the reference point, when µ T 1 (p wip w1 ) ⊥ tends to 0. It can be fixed by adding a second reference point P 2 if its corresponding image point p 2 is far enough from the singular line. Also, the relationship of the relative projective structure regarding P 1 and P 2 is given by: γi /γ1 = γ2 /γ1 • γi /γ2. In the right image Fig. 6, p 1 and p 2 correspond to the right and left bottom corner, respectively.

IV. OPTIMIZATIONS

In case of large motions or in presence of severe occlusions, the optical flow can be mistaken. The current section introduces some improvement strategies to overcome this kind of issues, by taking advantage of the approach presented above. The first optimization consists in combining several complementary motion fields, by using Tensor Voting framework to choose at each pixel, which one provides the most relevant displacement. The second one addresses the problem of initializing the optical flow.

A. Multi-flow Tensor Voting

Even if it provides a reliable solution in almost cases, the pyramidal approach of the Lucas and Kanade algorithm actually induced some issues. Due to the relative aperture size in highest levels of the pyramid, the original assumption consisting in a uniform velocity into the neighborhood Ω, is often violated close to the object boundaries. It implies a wrong initialization of lower levels, especially in presence of occlusions. Also, the pyramidal approach forbids to initialize the optical flow algorithm with a prior motion field. Again, the relative size of the aperture is too large in the highest level regarding to the norm of an initial down-sampled motion field. On an other hand, the original algorithm performed on only one level is well-adapted to be initialized by a prior motion field when the image motion is continuous, since the aperture size would be consistent with the norm of the initial motion at each pixel. So we propose to combine both the advantage of the pyramidal and non-pyramidal approaches.

Let consider the 4-D input space detailed section II-B2 containing n different motion fields. At each pixel location corresponds n velocity vectors. By performing the voting stage as described in II-B1, each tensor provides its relative position to all its neighbors that do not belong to the same image location. The final motion field is simply built by selecting, at each pixel, the velocity vector corresponding to the tensor with the higher surface saliency value.

B. Motion field filtering

The second optimization concerns initialization of optical flow as mentioned above. Assuming that camera displacement is continuous, we can simply use the result of our previous iteration as initialization for the current estimation of optical flow on one level, and combine it with pyramidal optical flow. The result is illustrated Fig. 8 with the bottom-left image, while the first input frame and the depth map computed without any optimization are given in the first row. Even if we can observed some improvements, especially in the bottom of the map, the depth is still not correctly evaluated in some large ground areas that are too close to image parts occluded due to the ego-motion, or not evaluated when motion field has been previously filtered due to discontinuities.

To initialize optical flow algorithm with a dense motion field, despite the threshold applied at the end of the Tensor Voting process, we propose to replace every missing velocity vectors with the corresponding homographic transformation induced by the ground plane under the horizon. We also proceed in the same manner above the horizon, by using the rotational homographic transformation. The result is shown in the bottom-right image Fig. 8: the road depth appears clearly better estimated up to the occluded area boundaries. Note that the depth map Fig. 6 includes these improvements and also presents a clear demarcation between the road and the occluded regions.

V. CONCLUSION

We presented a two-frames depth map estimation process, as well as its partial GPU integration that allows real-time constraints. The overall approach, with the optimization steps, is performed at about 12 frames per second for on midresolution images (640 × 480 pixels). The hardware configuration includes a 448 Cuda core architecture (GPU) combined with a dual core processor (CPU). It is easily scalable to the next generation of architecture to increase both speed and accuracy.

Because the planar parallax constraints are directly inferred by the epipolar ones, the depth maps we proposed are only relevant for static objects, and we plan to investigate the way to extend the 3-D recovering to mobile objects too. Also, we will focus on the integration on these results over more than two frames, and improve the optical flow filtering.

Figure 1 .

 1 Figure 1. (Left) synthetic Yosemite video sequence where the green arrow shows the ego-motion of the virtual camera. (Right) the average norm and angular error of the Lucas Kanade optical flow, relatively to |Ω ROI |.

Figure 2 .

 2 Figure 2. Results of the pyramidal approach of the Lucas and Kanade's optical flow algorithm, from the OpenCV implementation (left) and ours with CUDA (right). In addition of an execution 100 times faster, GPU allows to interpolate the motion field over-sampled at each level of the pyramid by hardware, avoiding the artifacts we can observe with the OpenCV version.

Figure 3 .

 3 Figure 3. Representation of the sub-space (x, y, vx) (right) corresponding to the optical flow field illustrated by the left image. In such a space, road, as well as the car coming in our direction or the background, describe different smooth surfaces, in the opposite of the wrong motion estimation.

Figure 4 .

 4 Figure 4. (Top) the average norm and angular motion field error relative to the surface saliency estimated by Tensor Voting, and the resulting optical flow after applying a threshold on the saliency value (bottom).

Figure 5 .

 5 Figure 5. The warping distance (top-right) is more densly consistent than the photometric one (top-left). Also, it is robust to thresholding as illustrated by the area of the road which appears stable despite the wide range of threshold values (bottom line).

Figure 6 .

 6 Figure 6. Relative projective structure of every image point, computed regarding only one reference point (left), and two reference points (right).

Figure 7 .

 7 Figure 7.

Figure 8 .

 8 Figure 8. Depth map computed with different level of optimization (see text for details).


Antoine Echelard

Jacques Lévy Véhel

Jacques Levy-Vehel

Claude Tricot

A Unified

Keywords: Multifractal analysis, 2-microlocal analysis, multifractal formalism, 2-microlocal formalism, wavelet analysis AMS Classication: 26A16, 28A80, 42C40

published or not. The documents may come

Introduction and Background

The analysis of global regularity, classically based on the global Hölder exponent, is adapted for the study of homogeneous signals. However, the global Hölder exponent yields insucient information when the regularity of a function evolves in time. Studying such functions requires tools that allow to characterize their behaviour at or around any point. One such tool is the pointwise Hölder exponent. We shall denote α p (f, x 0 ) the exponent of the function f at the point x 0 . Multifractal analysis [FP85, AP96, BMP92, CLP87, EM92, Fan97, HJK + 86, 1 KP76, Jaf97a, VT04, VV98, Man74, Ols95] studies the structure of the pointwise Hölder function, i.e. the function x 0 → α p (f, x 0 ): more precisely it aims at obtaining the multifractal spectrum, a function which measures the size of the level lines of α p (f, x). Both the theoretical and the numerical computations of this spectrum are dicult. This is why physicists and mathematicians have investigated a "multifractal formalism", which allows, in certain situations, to obtain the spectrum as the Legendre transform of a function that can be computed more easily.

Instead of focusing on the pointwise Hölder exponent and the ne structure of α p (f, x 0 ), one may follow a dierent approach and try to obtain a richer description of the local regularity at any xed point by means of other exponents, such as the local Hölder exponent [START_REF] Guiheneuf | 2-microlocal analysis and application in signal processing[END_REF], the chirp exponent [START_REF] Meyer | Wavelets, vibrations and scaling[END_REF], the oscillation exponent [START_REF] Arneodo | Singularity spectrum of multifractal functions involving oscillating singularities[END_REF] or the weak scaling exponent [START_REF] Meyer | Wavelets, vibrations and scaling[END_REF]. A powerful way to do so is to study the 2-microlocal frontier, dened in [START_REF] Guiheneuf | 2-microlocal analysis and application in signal processing[END_REF][START_REF] Meyer | Wavelets, vibrations and scaling[END_REF] based on the local version of the 2-microlocal spaces introduced by J.M. Bony in [START_REF] Bony | Propagation et interaction des singularités pour les solutions des équations aux dérivées partielles non-linéaires[END_REF]. The main interest of these spaces is that they allow to describe completely the evolution of the pointwise Hölder exponent at any given point under integro-dierentiation. The 2-microlocal frontier is a curve in an abstract space that is associated to each point, and that allows to predict this evolution. The 2-microlocal spaces were originally dened through a Littlewood-Paley decomposition. They were then characterized by conditions on the wavelet coefcients [START_REF] Jaard | Pointwise smoothness, 2-microlocalization and wavelet coecients[END_REF]. Time domain characterizations of increasing generality have been provided in [START_REF] Kolwankar | A time domain characterization of the ne local regularity of functions[END_REF][START_REF] Tricot | The 2-microlocal formalism. Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot[END_REF][START_REF] Echelard | Analyse 2-microlocale et application au débruitage[END_REF]. See also [START_REF] Meyer | Wavelets, vibrations and scaling[END_REF] for related results.

The computation of the 2-microlocal frontier is somewhat delicate. A 2microlocal formalism has been studied in [START_REF] Guiheneuf | 2-microlocal analysis and application in signal processing[END_REF][START_REF] Tricot | The 2-microlocal formalism. Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot[END_REF][START_REF] Echelard | Analyse 2-microlocale et application au débruitage[END_REF], with an approach that is analogous, in many respects, to the one of the multifractal formalism: at any xed point, the 2-microlocal frontier is the Legendre transform of a certain function called the 2-microlocal spectrum.

Thus, for a function f , the multifractal spectrum characterizes the level sets of the pointwise Hölder function, while the 2-microlocal spectrum allows to predict the change of regularity by integro-dierentiation at any point in the domain of f . These two descriptions yield a rather rich picture of the regularity, and they may be approached through related formalisms, which are essentially based on a Legendre transform.

In this work, we elaborate on the similitudes between the two formalisms.

We also study the problem of prescribing both the 2-microlocal and multifractal spectra. In the next section, we expose some general notions that are useful in both settings. In Section 3, we provide an abstract (weak) formalism. This formalism is applied to various versions of the multifractal spectra in Section 4, and to the 2-microlocal spectrum in Section 5. Finally, Section 6 presents results on the prescription of the spectra.

Sections 2 and 3 stay at a very general level. As a consequence, the denitions and results they present might appear rather abstract to the reader. However, as will be apparent in Sections 4 and 5, they contain the essence of what is common to the multifractal and 2-microlocal formalisms. In particular, propositions 4.1, 4.6, 5.2 and 5.3 may be seen as concrete examples of applications of this abstract formalism. In fact, the results in Sections 2 and 3 elucidate the very mechanisms relating the spectra, and might have applications in other settings.

2 Recalls: basic properties of functions dened on

P(R)
We recall in subsections 2.1, 2.2 and 2.3 some known denitions and results on set functions. In subsection 2.4, we specialize to a case which will be relevant for both 2-microlocal and multifractal analysis.

Common frame

We consider a function F :

P(R) → R = R ∪ {±∞}. F is called non-decreasing if for all real sets E 1 , E 2 , E 1 ⊂ E 2 ⇒ F (E 1 ) ≤ F (E 2 ).
It is called stable (with respect to union) if

F (E 1 ∪ E 2 ) = max{F (E 1 ), F (E 2 )}. Given a real k, F is k-stable if F (E 1 ∪ E 2 ) -k ≤ max{F (E 1 ), F (E 2 )} ≤ F (E 1 ∪ E 2 ) + k.
It is easily checked that any function G : P(R) → R that is non-decreasing and sub-additive (i.e. G(

E 1 ∪ E 2 ) ≤ G(E 1 ) + G(E 2 )) is such that F := log G is k-stable with k = log 2.

Non-decreasing F

Assume that F is non-decreasing. For all real α, the function F ([α -, α + ]) has a limit when tends to 0. One may then dene the localized function (also called max-plus limit).

F loc (α) = lim →0 F ([α -, α + ]). Lemma 2.1. Let F be non-decreasing. For all open set O in R, sup α∈O F loc (α) ≤ F (O).
(1) Proof. Indeed, if α ∈ O, there exists such that [α -, α + ] ⊂ O, and thus

F ([α -, α + ]) ≤ F (O). Lemma 2.2. If F is non-decreasing, then F loc is upper-semi-continuous.
Proof. The preceding lemma implies that, for > 0, sup β∈(α-,α+ ) F loc (β) ≤ F ((α -, α + )). Letting tend to 0, one gets the semi-continuity of F loc in α.

Lemma 2.3. If F is non-decreasing, then F loc reaches its supremum on any compact set of R.

Proof. This is a direct consequence of the semi-continuity.

Stability of F

If F is stable, then it is also non-decreasing: indeed, if E 1 ⊂ E 2 , F (E 1 ) ≤ max{F (E 1 ), F (E 2 )} = F (E 1 ∪ E 2 ) = F (E 2 ).
Lemma 2.4. Let F be stable. For any compact set K in R,

F (K) ≤ max α∈K F loc (α).
(2)

Proof. The proof uses the closed dyadic intervals of R. Since K is bounded, it may be covered by a nite number of dyadic interval of rank 0 (that is, of length 1). Since F is stable, one of these intervals, denoted J 0 , is such that F (J 0 ∩ K) = F (K). However, J 0 is covered by two intervals of rank 1, say J and J . One of these intervals, denoted J 1 , is also such that F (J 1 ∩ K) = F (K). By recurrence, one may construct a nested sequence (J n ) of dyadic intervals such that, for all n, F (J n ∩ K) = F (K).

Let α * denote the limit of the J n . Since K is closed, α * is a point of K. For any > 0, there exists an integer n such that J n ⊂ [α * -, α * + ]. Using the fact that F is non-decreasing, one gets that:

F (J n ) ≤ F ([α * -, α * + ]). Thus F (K) ≤ F ([α * -, α * + ]
), and, letting tend to 0:

F (K) ≤ F loc (α * ) . As a consequence, F (K) ≤ sup α∈K F loc (α).
Finally, since F is non-decreasing, F loc reaches its supremum on K.

Corollary 2.1. Let F be stable and K be a compact set with non-empty interior

• K. Then sup{F loc (α)/α ∈ • K} ≤ F (K) ≤ max{F loc (α)/α ∈ K}. (3)
Proof. This is a consequence of lemmas 2.1 and 2.4.

The relations (3) entail at once that, for any α and ,

sup |β-α|< F loc (β) ≤ F ([α -, α + ]) ≤ max |β-α|≤ F loc (β). (4)

A simple example

The following example aims at giving a straightforward application of the previous results, as well as at introducing some notations that will be used in the sequel.

Given a function g : R → R, one derives a set function by setting:

M (g, E) = sup t∈E g(t).
(5)

It is straightforward to check that M is stable. One may thus apply the preceding analysis. The corresponding localized function, denoted g, is welldened, and it is upper-semi-continuous:

g(α) = lim →0 M (g, [α -, α + ]). (6)
From (6), one easily checks that this is the smallest upper-semi-continuous function that is larger than g, i.e. its upper-semi-continuous envelope.

Finally, the inequalities (3) yield that, for any compact set K with non-empty interior:

sup{g(α)/α ∈ • K} ≤ sup t∈K g(t) ≤ sup{g(α)/α ∈ K}.
Note that symmetrical results may be obtained by setting:

m(g, E) = inf t∈E g(t). (7)
The lower-semi-continuous envelope of g is g(α) = lim →0 m(g, [α -, α + ]).

Since (-m) is stable, one gets the lower-semi-continuity of g and the bounds:

inf{g(α)/α ∈ K} ≤ inf t∈K g(t) ≤ inf{g(α)/α ∈ • K}.

A weak formalism

In this section, we consider a particular form for the function F . This form allows to write a general (weak) formalism, stated in lemma 3.1 and equality (12). Specializing this form allows to treat the cases of both the multifractal (Section 4) and 2-microlocal (Section 5) spectra. Indeed, all the applications in the sequel will deal with set functions P(R) → R which have a particular form that we now proceed to describe.

We assume given:

• A subset L of R n whose closure contains the origin 0.

For instance, one may consider L = (0, 1], L = {k2 -j , (j, k) ∈ Z 2 } or L = (0, 1] × (0, 1].

• A function h : L → R * + such that lim t→0 h(t) = 0.
For a function U :

L × P(R) → R (t, E) → U t (E)
we set:

F U (E) = lim sup t→0 h(t)U t (E) : t ∈ L . (8
U (t, E 1 ∪ E 2 ) -k ≤ max{U t (E 1 ), U t (E 2 )} ≤ U (t, E 1 ∪ E 2 ) + k.
Multiplying by h(t) and passing to the upper-limit gives the result.

The following result may be considered as an abstract weak formalism. It will be used to prove propositions 4.1, 4.6, 5.2 and 5.3: Lemma 3.1. Let k be a real number and U , V be such that U t , V t are k-stable for all t.

Assume there exist a function g : R → R and two positive constants c 1 and c 2 such that, for all t and all compact intervals I,

c 1 + U t (I) + m(g, I) h(t) ≤ V t (I) ≤ c 2 + U t (I) + M (g, I) h(t) . (9)
Then, for all t ∈ R,

F loc U (t) + g(t) ≤ F loc V (t) ≤ F loc U (t) + g(t). (10)
Furthermore, for all compact sets K with non-empty interior,

sup α∈ • K {F loc U (t) + g(t)} ≤ F V (K) ≤ sup t∈K {F loc U (t) + g(t)}. (11)
Finally, if g is continuous, then:

sup α∈R {g(α) + F loc U (α)} = lim R→∞ F V ([-R, R]). (12)
Proof. Multiplying (9) by h(t) and passing to the upper limit when t tends to 0, one gets:

F U (I) + m(g, I) ≤ F V (I) ≤ F U (I) + M (g, I). (13)
It then suces to replace I by [t -, t + ] and to let tend to 0 to obtain (10). Since F V is stable (by proposition 3.1), one may use (2) and (10) to get (11). Assume now that g is continuous. Set I = [-R, R] and let R tend to innity.

Then the inequalities (11) reduce to equality (12)

We end this section with a remark concerning equality (12): the non-decreasing [START_REF] Jaard | Multifractal formalism for functions II: selfsimilar functions[END_REF] and capacities [START_REF] Véhel | Multifractal analysis of Choquet capacities: preliminary results[END_REF], introducing more rened spectra [START_REF] Véhel | On various multifractal spectra[END_REF].

property ensures that F V (R) ≥ lim R F V ([-R, R]).
There is also a huge amount of work about estimating the multifractal spectra.

A very partial list of reference is [Can98, CJ92, Lév96, GL05, GL07, GH10, TPVG06].

As we recall below, there exists a variety of multifractal spectra. They contain dierent information, and some are easier to compute than others. The so-called strong multifractal formalism, when valid, asserts that the Hausdor multifractal spectrum is equal to the Legendre one (see below for denitions). This formalism will not be our concern here. Rather, we will concentrate on the weak multifractal formalism, which studies conditions under which the large deviation multifractal spectrum coincide with the Legendre one.

First introduced as a means to estimate the Hausdor spectrum, the large deviation multifractal spectrum has then proved a useful tool on its own, in particular in applications [START_REF] Véhel | Fractional Brownian motion and data trac modelling: the other end of the spectrum[END_REF][START_REF] Zähle | Multifractal analysis of Choquet capacities: Preliminary results[END_REF]). A disadvantage of the original denition of this spectrum is that it is based on an arbitrary partition of the support of the function to be analyzed. Thus, dierent partitions may in general lead to dierent spectra. To overcome this drawback, a continuous large deviation spectrum has been introduced in [START_REF] Véhel | On various multifractal spectra[END_REF]. We will in this work study both large deviation spectra.

The multifractal formalisms we will study will assert that, under some conditions, the discrete Legendre spectrum is the concave envelope of the original large deviation spectrum. Likewise, the continuous Legendre spectrum will be the concave envelope of the continuous large deviation spectrum

There is no added complication in considering an abstract set function A as a basis for dening the spectra instead of the usual coarse-grained exponents of a measure or real function. The correspondence between A and the quantities classically considered is as follows:

• For the analysis of a Borel measure µ, set A(u) = log µ(u)/ log |u|.

• For the analysis of a real function z, set A(u) = log v z (u)/ log |u|, where v z (u) is a measure of the variation of z in the interval u. Typical choices include the increments |z(u max ) -z(u min )| (where u = [u min , u max ]), the oscillation sup t∈u z(t)-inf t∈u z(t), or, when u is the dyadic interval [k2 -n , (k+ 1)2 -n ], the modulus of the wavelet coecient of z at scale n and location k (note that, for applications in signal processing, one uses slight modications known as the wavelet maxima or wavelet leader methods).

In this section, we shall apply the results of Section 2 to recover known results on the weak multifractal formalism, both in the discrete and continuous frames.

We shall also obtain some new results in the case where A(u) = log µ(u)/ log |u|, with µ an absolutely continuous measure.

Notations

X will denote the set of closed intervals of [0, 1]. We let X n be the set of intervals of the form [k2 -n , (k + 1)2

f h (α) = dim H (E(α)).
Of course, it is in general dicult to compute these Hausdor dimensions.

Most other multifractal spectra have been introduced as means to provide an indirect access to f h .

Discrete weak multifractal formalism 4.4.1 Discrete large deviation spectrum

From a heuristic point of view, the large deviation spectrum describes the asymptotic behaviour of the density of the intervals of X n having a coarse-grained Hölder exponent close to given value. More precisely, it is dened as follows.

Let A : X → R + be a measurable function.

For any measurable subset E of R, let N (n, E) denote the number of intervals I of X n such that A(I) belongs to E:

N (n, E) = #{I ∈ X n such that A(I) ∈ E}.
In order to assess the behaviour of N (n, E) when n tends to innity, we consider the function

F d g (E) = lim sup n→∞ log N (n, E) n log 2
and its localized version

f d g (α) = lim →0 F d g ([α -, α + ]).
This function is called the discrete large deviation spectrum associated to

A [Fal03, LV98].
Roughly speaking, if for all > 0, the number of intervals I of X n for which

A(I) belongs to [α 0 -, α 0 + ] is proportional to 2 r.n , then f d g (α 0 ) = r. Another
interpretation, that provides a link with the continuous spectrum to be introduced below, is as follows: if, for all > 0, the union of dyadic intervals I of length 2 -n for which A(I) belongs to [α 0 -, α 0 + ] has measure proportional to 2 -n .2 r.n , then f d g (α 0 ) = r.

Discrete Legendre spectrum

Dene, for all q ∈ R:

S q (n) = {2 -nqA(I) : I ∈ X n and A(I) < ∞} and τ d (q) = lim inf n→∞ log S q (n) -n log 2 .
We shall use in the sequel the following quantities:

S q (n, E) = {2 -nqA(I) : I ∈ X n and A(I) ∈ E}.
f d l (α) = sup q∈R {qα -τ d (q)}.

Application of Section 3

We apply the results of Section 3 to the quantities dened in Sections 4.4.1 and 4.4.2. This will yield the weak multifractal formalism. More precisely, we set:

L = {2 -n , n ∈ N}, h(2 -n ) = 1/(n log 2), U 2 -n (E) = log N (n, E).
Clearly, E → N (n, E) is non-decreasing and sub-additive. U 2 -n is thus kstable for all n, with k = log(2). It is easy to check that, with the notations of Section 3:

F U = F d g and F loc U = f d g .
Set:

V 2 -n (E) = log {2 -nqA(I) : I ∈ X n and A(I) ∈ E}.

Now, E → {2 -nqA(I) : I ∈ X n and A(I) ∈ E} is non-decreasing and subadditive, thus V 2 -n is k-stable for all n with k = log(2).

Furthermore, V 2 -n may be written as:

V 2 -n (E) = log S q (n, E),
and thus:

τ d (q) = -F V (R + ).
Set for convenience:

τ d R (q) = -F V ([0, R]).
We wish to apply lemma 3.1. In that view, remark that:

N (n, E) inf α∈E 2 -nqα ≤ {2 -nqA(I) : I ∈ X n and A(I) ∈ E} ≤ N (n, E) sup α∈E 2 -nqα . ( 14
)
Set g(α) = -q α. Taking logarithms, the inequalities (14) read

U 2 -n (E) + inf α∈E g(t) h(2 -n ) ≤ V 2 -n (E) ≤ U 2 -n (E) + sup α∈E g(t) h(2 -n ) .
Lemma 3.1 applies. Writing relation (12) in this frame, one nds:

Proposition 4.1. For all function A : X → R + and q ∈ R,

inf α∈R {αq -f d g (α)} = lim R→∞ τ d R (q). (15)
Recall that, since F V is non-decreasing, lim R→∞ τ d R (q) ≥ τ d (q). Let us now nd a condition implying equality of these two terms. First, note that the equality is always true for q > 0. Indeed: Proposition 4.2. For all function A : X → R + and all q > 0 inf

α∈R {αq -f d g (α)} = τ d (q). (16)
Proof. For any natural number n, there exists 2 n intervals of order n, thus at most 2 n dyadic intervals I of order n such that A(I) > R. As a consequence,

S q (n) ≤ S q (n, [0, R]) + 2 n 2 -qR . ( 17
)
For any sequences (a n ) and (b n ) of positive reals,

lim sup log(a n + b n ) n = max{lim sup log a n n , lim sup log b n n }.
Set a n = S q (n, [0, R]) and b n = 2 n(1-qR) . One deduces:

lim sup n→∞ log S q (n) n log 2 ≤ max lim sup n→∞ log S q (n, [0, R]) n log 2 , 1 -qR
and thus:

min{τ d R (q), qR -1} ≤ τ d (q) for all R.
When R tends to +∞, so does qR -1 (this is where the assumption q > 0 is used), which entails τ d (q) = lim R→∞ τ d R (q).

An additional assumption is necessary for the equality to hold when q ≤ 0 (see Section 4.6 for a counter-example). For instance, the result is valid if A(I) is bounded for |I| small enough: Proposition 4.3. Assume there exist η > 0 and M ∈ R such that, for all dyadic interval I, |I| ≤ η implies that A(I) ≤ M or A(I) = +∞. Then, for all q ∈ R: When the assumption of the previous proposition does not hold, the value of τ d (q) for q < 0 is known. Indeed: Proposition 4.4. If, for all η > 0 and M ∈ R, there exists a dyadic interval I such that |I| ≤ η and M < A(I) < +∞, then for all q < 0:

inf α∈R {αq -f d g (α)} = τ d (q). Proof. For all R ≥ M : S q (n, [0, R]) = S q (n) as soon as 2 -n ≤ η; thus τ d R (q) = τ d (q).
τ d (q) = -∞.
Proof. For all M > 0, there exists a subsequence I σ(n) such that M < A σ(n) < +∞. This implies that S q (σ(n)) ≥ 2 -qσ(n)M , and thus that log S q (σ(n))

-σ(n) log 2 ≤ qM . Taking lim inf, one gets τ d (q) ≤ qM .

Cases where A is not bounded in the neighbourhood of |I| = 0 and where τ d R (q) does not tend to -∞ for q < 0 do exist. We shall provide an example in the next section, in the continuous frame.

An important particular case is when A may be written as A = log µ(I)/ log |I|, where µ is a positive measure. Note that, in this case, by denition of τ , τ d (0) = -dim B (support(µ)), where dim B stands for box dimension.

Theorem 4.1. Let A(I) = log µ(I) log |I| , where µ is a positive measure non identically zero. Then, for all q = 0:

τ d (q) = inf α∈R {qα -f d g (α)}.
The proof of the theorem will use the following lemma:

Lemma 4.1. Let q < 0. If, for all r > 0, there exists R 0 > 0 such that: for all N ∈ N, there exist n ≥ N and I ∈ X n such that A(I) ∈ [r, R 0 ], then:

lim R→∞ τ d R (q) = -∞
Proof. For all R > R 0 and all N ∈ N, there exist n ≥ N and I ∈ X n such that

A(I) ∈ [r, R].
As a consequence, S q (n, [0, R]) ≥ 2 -nqr , and thus:

log S q (n, [0, R]) -n log 2 ≤ qr
Taking lim inf on n, one gets:

τ d R (q) ≤ qr,
and the result follows.

Proof of theorem 4.1. In view of propositions 4.1, 4.3 and 4.4, it is enough to prove that, when A is not bounded in the neighbourhood of 0, one has, for all q<0:

lim R→∞ τ d R (q) = -∞.
Assume then that A is not bounded in the neighbourhood of |I| = 0: for all η > 0 and M ∈ R, there exists a dyadic I with |I| ≤ η such that M < A(I) < +∞.

We shall show that this implies that the assumptions of lemma 4.1 are veried.

Assume the opposite: then there exists r > 0 such that, for all R 0 > 0, there exists N ∈ N such that, for all n ≥ N and all I ∈ X n :

A(I) / ∈ [r, R 0 ]. This implies: µ(I) ≤ 2 -nR 0 or µ(I) ≥ 2 -nr . ( 18
)
Choose R 0 = max(2r, 2). Since A is unbounded in the neighbourhood of |I| = 0, there exists a dyadic |I| of order n ≥ 2 such that R 0 < A(I) < +∞, i.e.

0 < µ(I) ≤ 2 -nR 0 .
One may divide I into two dyadic intervals I 1 and I 2 of order n + 1. Since I 1 ⊂ I, µ(I 1 ) ≤ µ(I), and thus µ(I 1 ) ≤ 2 -nR 0 . However, R 0 .n > r.(n + 1) which entails µ(I 1 ) < 2 -(n+1)r . From (18), one deduces:

µ(I 1 ) ≤ 2 -(n+1)R 0 . Likewise, µ(I 2 ) ≤ 2 -(n+1)R 0 .
An obvious induction shows that, for any dyadic interval J of order n + m, m ≥ 0, included in I: µ(J) ≤ 2 -(n+m)R 0 . Thus, µ(I) ≤ 2 m-(n+m)R 0 . Since R 0 > 1, one gets, by letting m tend m to innity, µ(I) = 0, which contradicts the assumption.

The case q = 0 appears to be more dicult to deal with. If one assumes in addition to the above that µ is absolutely continuous, then the following results hold: Proposition 4.5. Assume A(I) = log µ(I) log |I| , where µ is an absolutely continuous positive measure non identically zero. Then:

τ d (0) = inf α∈R {-f d g (α)} = -1.
As a consequence, for all q ∈ R:

τ d (q) = inf α∈R {qα -f d g (α)}.
Proof. Let us rst compute the value of τ d (0):

By denition, τ d (0) = -dim(support(µ)). Since µ is absolutely continuous and non zero, τ d (0) = -1.

Let us now evaluate the limit of τ R (0) when R tends to innity. The assumptions on µ entail that there exists an integrable positive function such that µ(I) = I f (x)dx. Since µ is not zero, there exist > 0 and a bounded Borel set B of positive measure such that f (x) ≥ .1 x∈B . Set

ν(I) = I 1 x∈B dx.
Since ν ≤ µ, for any interval I with diameter smaller than 1, we deduce that

A(I) ≤ log ν(I) log |I| . Set u n = I ∈ X n : ν(I) ≥ 2 -n |B| 2 .
We have |B| = {ν(I) : I ∈ X n }. Since, for all I ∈ X n , ν(I) ≤ 2 -n , we obtain: |B| ≤ 2 -n u n + 1 2 2 -n |B| (2 n -u n ) and thus:

u n ≥ 2 n K
where K = |B|/2 is strictly positive.

As a consequence,

I ∈ X n : A(I) ≤ 1 + log |B| 2 -n log 2 ≥ 2 n K. Choose n large enough so that log |B| 2 -n log 2 ≤ 1: {I ∈ X n : A(I) ∈ [0, 2]} ≥ 2 n K and thus: S 0 (n, [0, 2]) ≥ 2 n K
Taking logarithms and passing to the lim inf, one gets: τ d 2 (0) ≤ -1. The result follows since R → τ d R (0) is not increasing with upper limit -1.

The second part of the proposition follows from proposition 4.2.

Continuous Multifractal Formalism

Continuous Large Deviation Spectrum

The very denition of the large deviation spectrum entails that it depends on a choice of a partition of the support of the signal (usually the dyadic intervals). This is a drawback, since, in general, the theoretical spectrum may indeed be dierent with dierent choices of partitions for the same signal. Even in the case where the theoretical spectrum does not depend on the partition, numerical estimation will typically yield dierent results with dierent choices of intervals.

The continuous large deviation spectrum was introduced in [VT04] as a means to dene a partition-free spectrum. The idea is to consider all sets of a given length η instead of restricting to dyadic intervals. The continuous large deviation spectrum then measures the asymptotic behaviour, when η tends to 0, of the density of the union of intervals of size η having a coarse-grained exponent close to a given value.

Even though going from a discrete setting to a continuous one may appear as a complication, it actually eases the estimation of the large deviation spectrum, by yielding more robust results on numerical data. This is similar to the dichotomy discrete vs continuous wavelet transform: it is well-known that the continuous wavelet transform provides better results as far as estimation is concerned (at the expense of an additional computational burden). As in the continuous wavelet case, the numerical computation of the continuous spectrum is performed by considering all samples at each scale, instead of using an averaging or sub-sampling as in the usual case.

As above, we consider a general set function A instead of specic coarsegrained exponents.

As compared to the discrete case, we replace the sets X n by the sets R η , dened, for all η ∈ (0, 1), by: R η = {I ∈ X : |I| = η}.

Recall that, for all set of intervals T , we denote ∪T = ∪{I : I ∈ T }.

For any subset T of R η and any t ∈ ∪T , the largest interval containing t and included in ∪T has length not smaller than η. In addition, these intervals are disjoint. As a consequence, there are nitely many of them. This entails that ∪T is measurable. This allows to dene the equivalent of N (η, E):

N c (η, E) = 1 η |∪{I ∈ R η : A(I) ∈ E} |.
Similarly to the discrete case, we seek to characterize the behavior of N c (η, E) when η → 0. In that view, consider

F c g (E) = lim sup η→0 log N c (η, E) -log η .
The continuous large deviation spectrum associated to A is dened as:

f c g (α) = lim →0 F c g ([α -, α + ]).
Roughly speaking, if the union of the intervals I of length η for which A(I) belongs to [α 0 -, α 0 + ] has measure proportional to η.η -r.n for all suciently small , then f d g (α 0 ) = r.

Continuous Legendre Multifractal Spectrum

To adapt the quantities S q dened in the discrete case in this setting, one uses the notion of packing [START_REF] Véhel | On various multifractal spectra[END_REF]: a family R ⊂ X is a packing of [0, 1] if the intervals of R have pairwise disjoint interiors. One then dene, for any R ⊂ X :

H q (R) = sup {|u| qA(u) : u ∈ R and A(u) < +∞} : R is a packing of [0, 1] drawn from R (19)
with the conventions:

|u| +∞ = 0, |u| -∞ = +∞, H q (∅) = 0. Set τ c (q) = lim inf η→0 log H q (R η ) log η .
The continuous Legendre spectrum associated to A is the Legendre transform of τ c : L c (α) = sup q∈R {qα -τ c (q)}.

Remark that, when R is a packing, H q (R) may be written

H q (R) = |u| qA(u) : u ∈ R and A(u) < +∞ .
This shows that H q is indeed a generalization of S q : X n is a packing, thus H q (X n ) = |u| qA(u) : u ∈ X n and A(u) < +∞ = S q (n).

Application of Section 3

The results of Section 3 apply to the quantities dened in Section 4.4.1 in a continuous frame with the following correspondences: E) is non-increasing and sub-additive. This means that U η is k-stable with k = log 2 for all η.

L = (0, 1], h(η) = 1/| log η|, U η (E) = log N c (η, E). Remark that, for any disjoint E 1 , E 2 , N c (η, E 1 ∪ E 2 ) = N c (E 1 ) ∪ N c (E 2 ). As a consequence, E → N c (η,
One easily checks that, with the notations of Section 3:

F U = F c
g and F loc U = f c g .

Set:

V η (E) = log H q (R η ∩ A -1 (E)).
where H q is dened in Section 4.5.2. We note that, for all disjoint

E 1 , E 2 , H q (R η ∩ A -1 (E 1 ∪ E 2 )) = H q (R η ∩ A -1 (E 1 )) + H q (R η ∩ A -1 (E 2 )). This entails that E → H q (R η ∩ A -1 (E)
) is non-decreasing and sub-additive. Thus, V η is k-stable with k = log 2. Furthermore,

τ c (q) = -F V (R + ).
Set for convenience:

τ c R (q) = -F V ([0, R]) = lim inf η→0 log H q (R η ∩ A -1 ([0, R])) log η .
We wish to apply lemma 3.1. In that view, we shall make use of the following result from [START_REF] Véhel | On various multifractal spectra[END_REF], whose proof we recall for completeness:

For all E ∈ P(R):

1 2 N c (η, E) inf α∈E η qα ≤ H q (R η ∩ A -1 (E)) ≤ N c (η, E) sup α∈E η qα . ( 20
)
Proof. Set T = R η ∩ A -1 (E).

We start with the right-hand side inequality. Let E ∈ P(R). For any packing R drawn from T , one may write:

{η qA(u) : u ∈ R} ≤ sup u∈R {η qA(u) } R. However, R ≤ η -1 |∪T | ≤ N c (η, E), and, for all u ∈ R, A(u) ∈ E. Thus sup u∈R {η qA(u) } ≤ sup α∈E η qα ,
which was to be proved. Now, ∪T is a union of intervals with same length, and one may write ∪T as: ∪T = ∪ n k=1 J k where the J k are non-empty, closed, and disjoint. We show that, for all J k , one may extract from

T a packing R k such that η R k ≥ |J k |/2 and ∪R k ⊂ J k .
Indeed, let p be such that |J k | ∈ (2pη, 2(p + 1)η]. Choose a 0 ...a p in J k such that a q > a q-1 + 2η for 1 ≤ q ≤ p. We know that, for 0 ≤ q ≤ p, there exists an interval L q ∈ T such that a q ∈ L q . It is easily checked that the L q are disjoint and included in

J k . Set R k = {L q , q = 0..p}. Since R k = (p + 1), one has η R k ≥ |J k |/2. Furthermore, ∪R k ⊂ J k .
Consider now the union R of R k for k = 1..n: it is easy to see that R is a packing of [0, 1] drawn from T , whose cardinal is not smaller than |T |/2η. Since inf u∈R {η qA(u) } R ≤ {η qA(u) : u ∈ R}, one deduces:

1 2 |T | η inf u∈R {η qA(u) } ≤ {η qA(u) : u ∈ R}.
Set g(α) = -q α. Taking logarithms, the inequalities (20) read:

-

ln(2) + U η (E) + inf α∈E g(t) h(η) ≤ V η (E) ≤ U η (E) + sup α∈E g(t) h(η) .
Lemma 3.1 applies. Writing relation (12) in this frame, one nds:

Proposition 4.6. For any function A : X → R + and all q ∈ R,

inf α∈R {αq -f c g (α)} = lim R→∞ τ c R (q). ( 21
)
Recall that, since F V is non-decreasing, lim R→∞ τ c R (q) ≤ τ c (q). We look now for conditions that guarantee equality of these two terms. It is always veried when q > 0 :

Proposition 4.7. For any function A : X → R

+ and for all q > 0 inf

α∈R {αq -f c g (α)} = τ d (q). (22)
Proof. For all η > 0 and for all packing R drawn from R η , R ≤ η -1 . This entails:

H q (R) ≤ H q (R η ∩ A -1 ([0, R])) + 2 n 2 -qR . ( 23
)
Taking logarithms, dividing by log η and passing to the lim sup when η → 0:

min{τ c R (q), qR -1} ≤ τ c (q)
for all R. When R tends to +∞ so does qR -1 (recall that q > 0), and thus τ c (q) = lim R→∞ τ c R (q).

Additional assumptions on A are necessary for (22) to hold for all q. One may for instance require that A is bounded for suciently small |I|: Proposition 4.8. Assume there exists η 0 > 0 such that sup{A(I) :

|I| ≤ η 0 } < +∞. Then inf α∈R {αq -f c g (α)} = τ (q) for all q.
This result obviously corresponds to the one in proposition 4.3. Again, one easily checks that it applies in the case of multinomial measures, i.e. the weak multifractal formalism for continuous spectra holds for such measures.

When the assumption of the previous proposition does not hold, we do know the value of τ c (q) for q < 0: Proposition 4.9. If, for all η > 0 and M ∈ R there exists an interval I such that |I| ≤ η and A(I) > M , then for all q < 0:

τ c (q) = -∞.
Proof. For all M > 0, there exists a sequence I n such that A(I n ) > M and

|I n | → 0. As a consequence, H q (R |In| ) ≥ 2 -qnM , and thus log H q (R |In| ) -log |In| ≤ qM .
Taking lim inf, τ d (q) ≤ qM .

There exists cases where A is not bounded for small |I| and where lim R→∞ τ c R (q) = -∞ for q < 0.

Example Dene the function

A on all closed intervals of [0, 1] of positive length by A([a, b]) = 1 for a > 0 1/(b -a) for a = 0. Let q ∈ R and x R > 1.
For all η < 1/R , for all interval I with length η: A(I) ∈ [0, R] if and only if 0 / ∈ I. In that case, A(I) = 1.

A packing of intervals with length η contains at most η -1 intervals. As a consequence,

H q (R η ∩ A -1 ([0, R])) ≤ η q-1 .
Besides, the packing of intervals {[kη, (k + 1)η] : k = 1.. η -1 -1} is a packing of intervals of [0, 1] with cardinal η -1 -1 which does not contain 0. This implies that:

( η -1 -1)η q ≤ H q (R η ∩ A -1 ([0, R])).
Finally, we get:

τ c R (q) = q -1.
The sequence A([0, 1/n]) tends to +∞. Proposition 4.9 then entails that, for all q < 0:

τ c (q) = -∞.
We thus have an example where (22) does not hold for all q.

Remark that the sequence A([0, 2 -n ]) tends to +∞. From proposition 4.4, we get: τ d (q) = -∞. However, τ R d (q) ≥ τ R c (q) ≥ q -1: this provides thus a case where (16) does not hold either for all q.

More can be said under usual assumptions on A: Proposition 4.10. Let A(I) = log µ(I) log |I| where µ is a positive measure. Then inf α∈R {αq -f c g (α)} = τ (q) for all q = 0.

Proof. When the assumptions of proposition 4.7 or of proposition 4.8 are in force, then the desired equality holds.

We still have to deal with the more delicate case where q < 0 and for all η 0 > 0: sup{A(I) : |I| ≤ η 0 } = +∞.

We know from proposition 4.9 that, in this case, τ c (q) = -∞. We would thus like to show that lim R τ c R (q) = -∞. This is in fact a consequence of theorem 4.1: We assume given an admissible wavelet ψ with N vanishing moments and r rst derivatives having fast decay. We also x a real number x 0 . As usual, we denote:

just remark that τ c R (q) ≤ τ d R (q).
ψ (a,b) (x) = 1 √ a ψ x -b a (24)
the dilated and translated versions of ψ.

The 2-microlocal frontier of the function f at x 0 is characterized as follows: for all couple (σ, s ) ∈ R 2 such that r + s + inf(s , 1) > 0 and N > max(s, s + s ),

σ ≤ sv (s ) is equivalent to: σ ≤ lim inf (a,b)→0 s log(a + |b|) + log f, a -1/2 ψ (a,b+x 0 )
log a .

One deduces that, if r + sv (s ) -s + inf(s , 1) > 0 and N > max(sv (s ) -s , sv ),

then sv (s ) = lim inf (a,b)→0 s log(a + |b|) + log f, a -1/2 ψ (a,b+x 0 ) log a . ( 25
)
where sv is the 2-microlocal frontier of f at x 0 . Strictly speaking, one should dene sv ψ (s ) as being equal to the right-hand side term of equation ( 25) and remark that sv ψ and sv coincide on a certain subset of the plane. However, we shall in the sequel assume that ψ is suciently regular (i.e. that r and N are suciently large) so that, for all relevant s , (25) is veried.

The continuous 2-microlocal spectrum is dened as:

χ(ρ, ) = lim inf (a,b)→0 log f, a -1/2 ψ (a,b+x 0 ) log a : (a, b) ∈ D w (ρ, )
where

D w (ρ, ) = {(a, b) ∈ (-1, 1) 2 , such that a ρ+ ≤ max(a, |b|) ≤ a ρ-}.
For small enough, one may give an explicit description of the set D w (ρ, ) as follows:

• For ρ ∈ (0, 1): (a, b) ∈ D w (ρ, ) ⇐⇒ a ρ+ ≤ |b| ≤ a ρ-.

• For ρ = 0 : (a, b) ∈ D w (ρ, ) ⇐⇒ a ρ+ ≤ |b|.

• For ρ = 1 : (a, b) ∈ D w (ρ, ) ⇐⇒ |b| ≤ a ρ-.
The 2-microlocal spectrum of f at x 0 is dened by When ρ ∈ (0, 1),

χ(ρ) = lim →0 χ(ρ, ).
d(ρ) = (-1, 0] × {0} ∪ {(a, a 1/ρ )/a ∈ (0, 1)}
is the graph of a continuous function.

Dene then, for E ∈ P(R), the subset D E of (0, 1] × R:

D E = ρ∈E d(ρ). (26)
It is easy to check that D [ρ-,ρ+ ] = D w (ρ, ).

We make a nal remark that χ(ρ, ) may be written as follows:

χ(ρ, ) = lim inf (a,b)→0 a>0 b∈R log f, a -1/2 ψ (a,b+x 0 ) .1 (a,b)∈D w (ρ, ) log a ,
with the convention log 0 = -∞.

The next elementary inequalities will be useful in the sequel: Proof. For all (a, b) ∈ D w (ρ, ):

a ρ+ ≤ max(a, |b|) ≤ a + |b| ≤ 2 max(a, |b|) ≤ 2a ρ-.
Putting to the s power the inequalities a ρ+ ≤ a + |b| ≤ 2a ρ-yields the result.

Application of Section 3

The results of Section 3 apply with:

L = (0, 1] × R, t = (a, b), h(a, b) = 1 | log a| , U ((a, b), E) = log(| f, a -1/2 ψ (a,b+x 0 ) |.1 (a,b)∈D E ), V ((a, b), E) = log(| f, a -1/2 ψ (a,b+x 0 ) |.(a + |b|) s .1 (a,b)∈D E ).
The functions U and V are stable. F U ([ρ -, ρ + ]) corresponds to -χ (ρ), and F loc U (ρ) to -χ(ρ). Furthermore, F V ([0, 1]) = F V (R) = -σ(s ). Let g(t) = -s t. Proposition 5.1 may be written as:

∀(a, b) ∈ (0, 1] × [-1, 1]: -s + U (a, b, I) + m(g, I) h(t) ≤ V (a, b, I) ≤ s + U (a, b, I) + M (g, I) h(t) .
This implies that U and V fulll condition (9) with g(ρ) = -ρs . We may thus write equality (12) in this context, noting that χ(ρ) is equal to +∞ when ρ / ∈ [0, 1] and that -σ(s

) = F V ([0, 1]) = lim R→∞ F V ([-R, R]): max ρ∈[0,1] {-χ(ρ) -ρs } = -σ(s ).
One nally reaches the following result: Proposition 5.2. For all s ∈ R:

σ(s ) = min ρ∈[0,1] {χ(ρ) + ρs }.
This is the wavelet version of the 2-microlocal formalism, as exposed in [START_REF] Tricot | The 2-microlocal formalism. Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot[END_REF].

It allows for an easy computation of the 2-microlocal frontier of many classical functions including for instance the Weierstrass function or fractal interpolation functions.

Discrete 2-microlocal Formalism

Classical Formalism

We shall make use of the following notation:

ψ j,k (x) = ψ (2 -j ,k2 -j ) (x) = 2 j/2 ψ(2 j x -k). ( 27
)
The discrete 2-microlocal formalism is based on the discrete wavelet characterization of the 2-microlocal spaces given in [START_REF] Jaard | Pointwise smoothness, 2-microlocalization and wavelet coecients[END_REF]. As before, we assume that ψ is suciently regular so that, for all relevant values of s :

sv (s ) = lim inf (2 -j ,k2 -j -x 0 )→0 log 2 (| f, 2 -j/2 ψ jk |.(2 -j + |k2 -j -x 0 |) s -j . (28)
In order to get an expression that looks more like (25), we set:

L d (x 0 ) = {2 -j , k2 -j -x 0 , j > 0 , k ∈ Z}.
Equation (28) may be written as:

σ(s ) = lim inf (a,b)→0 log f, a 1/2 ψ (a,b+x 0 ) .(a + |b -x 0 |) s log a : (a, b) ∈ L d (x 0 ) . (29)
As above, we dene:

χ d (ρ, ) = lim inf (a,b)→0 (a,b)∈L d (x 0 ) log f, a 1/2 ψ (a,b+x 0 ) .1 (a,b)∈D [ρ-,ρ+ ]
log a , and

χ d (ρ) = lim →0 χ d (ρ, ).
As compared to the continuous 2-microlocal spectrum, the discrete one is less robust, since its computation involves a fewer number of wavelet coecients.

However, it may be prescribed more easily, since the f, ψ jk may be chosen as desired. The continuous 2-microlocal spectrum should then be more adapted for evaluating the frontier of a function at a given point, while the discrete one would be more tted in situations where one wishes to obtain a given frontier (this is for instance the case in applications such as signal denoising, see [START_REF] Echelard | Analyse 2-microlocale et application au débruitage[END_REF]).

Application of Section 3

As was the case for the continuous spectrum, the results of Section 3 apply.

The same correspondences may be used, with the exception that we set here L = L d (x 0 ). One then easily gets the following result:

Proposition 5.3. For all s ∈ R:

σ d (s ) = min ρ∈[0,1] {χ d (ρ) + ρs }.
Again, this formalism yields an easy way to compute the 2-microlocal frontier of functions such as fractal interpolation functions.

6 Prescribing Multifractal and 2-microlocal Spectra

As another example of the profound similarities between the multifractal and 2-microlocal formalism, we consider in this section the problem of prescribing the spectra in the discrete frame. Both the 2-microlocal and the multifractal spectra may be written in the form studied in Section 3: F U (E) = lim sup t→0 h(t)U (t, E).

As a consequence, the prescription of both spectra share some common features.

However, as we will see, the constraints on the function U are not the same in the two cases.

General Frame

This section gives a result that be will useful for the prescription of both spectra.

The following distance over R will be convenient in order to place ourselves in the most general frame: d R (x, y) = |arctan x -arctan y|, with the convention arctan(±∞) = ±π/2. It is easy to check that d R and the usual distance are topologically equivalent on R. Another useful remark is that d R (x, y) ≤ |x -y|.

The heuristic idea is as follows: assume we wish to construct a function whose spectrum is g on an interval I. We give ourselves a sequence I n of intervals that will "rotate" on I, and whose diameter will tend to 0. On each I n , we approximate g by its maximum over I n . This will be meaningful whenever g is upper semicontinuous. For a xed interval [α -, α + ], assumption (30) below ensures that there exists a subsequence of (I n ) n made of intervals contained in [α -, α + ], and for which x n V n (I n ) will be larger than g(α). Here (x n ) n is a positive sequence tending to 0. This entails F loc (α) ≥ g(α).

For the reverse inequality, one needs to ensure that the values of V n on intervals other than I n will not aect F . Assumption (31) says that one may choose n 0 such that I n 0 ⊂ [α -, α + ] and that when n → ∞, x n V n (I n 0 ) will be not larger than M (g, [α -2 , α + 2 ]). Lemma 6.1. Let g be an upper semi-continuous function on the interval I. Let (x n ) n be a positive sequence tending to 0, and let V n be k-stable for all n ∈ N.

Dene F as: F (E) = lim sup n→∞ x n .V n (E) for all E ⊂ I.

Let (I n ) n be a sequence of open intervals such that:

∀N ∈ N : I ⊂ ∪ n≥N I n , lim n→∞ |I n | = 0.
Assume that:

lim n→∞ d R (x n V n (I n ), M (g, I ∩ I n ) = 0. (30)
and that, for all n 0 :

lim sup n→∞ x n V n (I n 0 \ I n ) ≤ M (g, I n 0 ∩ I). (31)
Then F loc = g.

Proof. The assumptions on (I n ) n entail that, for all α ∈ I, there exists a subsequence I σ(n) such that, for all n, α ∈ I σ(n) . As g is upper semi-continuous, M (g, I ∩ I σ(n) ) tends to g(α) in R. Thus, by assumption, x σ(n) V σ(n) (I σ(n) ) tends to g(α) in R.

For all

> 0 and for all n such that |I σ(n) | < , the k-stability of V σ(n) entails that:

x σ(n) V σ(n) ([α -, α + ]) + k.x σ(n) ≥ x σ(n) V σ(n) (I σ(n) ).
Taking the lim sup when n tends to innity, one gets F ([α -, α + ]) ≥ g(α).

This inequality is true for all > 0, and thus F (α) ≥ g(α). We show now that F (α) ≤ g(α). For all n, I n 0 = I n 0 \ I σ 1 (n) . Thus, by assumption:

lim sup n→∞ x σ 1 (n) V σ 1 (n) (I n 0 ) ≤ M (g, I ∩ I n 0 ). Besides, V σ 2 (n) (I n 0 ) ≤ 2k + max(V σ 2 (n) (I n 0 \ I σ 2 (n) ), V σ 2 (n) (I σ 2 (n) )) and thus lim sup n→∞ x σ 2 (n) V σ 2 (n) (I n 0 ) ≤ max M (g, I ∩ I n 0 ), lim sup n→∞ (x σ 2 (n) V σ 2 (n) (I σ 2 (n) )) .
However, for n large enough, I σ 2 (n) ⊂ [α -2 , α + 2 ]. As a consequence, lim sup

n→∞ x σ 2 (n) V σ 2 (n) (I σ 2 (n) ) ≤ M (g, I ∩ [α -2 , α + 2 ]).
The lim sup of both sequences I σ 1 and I σ 2 are not larger than M (g, I ∩[α -2 , α + 2 ]) and thus F (I n 0 ) ≤ M (I ∩ [α -2 , α + 2 ]).

Since F is not decreasing, F (α) ≤ M (I ∩ [α -2 , α + 2 ]). Finally, let tend to 0 to obtain the result.

Remark The shape of the function F in lemma 6.1 obviously ts in the frame of Section 3: set for instance L = {1/n, n ∈ N}, h(1/n) = x n and U (1/n, I) = V n (I).

Remark For any interval I, it is easy to construct a sequence I n such as the one described in theorem 6.1. If, for instance, I = [0, +∞], then, for k an integer between 0 and +∞, one may divide the intervals (-1/k, k) into sub-intervals of the form ((i -1)/k, (i + 1)/k), where i is an integer varying from 0 to k 2 -1: I 1 = (-1, 1) I 2 = (-1/2, 1/2) I 3 = (0, 1) I 4 = (1/2, 3/2) I 5 = (1, 2) I 6 = (-1/3, 1/3) I 7 = (0, 2/3] I 8 = (1/3, 1)

• • • I 14 = (7/3, 3) etc...

Multifractal Spectrum

The prescription of the Hausdor spectrum has been considered in [START_REF] Véhel | Multifractal analysis of Choquet capacities: preliminary results[END_REF] et [START_REF] Jaard | Exposants de Hölder en des points donnés et coecients d'ondelettes[END_REF].

[VT04] has studied the prescription of various other related multifractal spectra.

We focus here on the prescription of the discrete large deviation spectrum.

In a discrete frame, one is concerned with the values of the function A only for intervals of the form: [(k -1)2 -n , k2 -n ] ; n ∈ N , k ∈ 1..2 j . To simplify notations, we shall write:

A nk = A([(k -1)2 -n , k2 -n ]).

By abuse of language, we shall speak of the multifractal spectrum of the A nk in place of the multifractal spectrum of the measure or function whose associated intervals function A is such that A([(k -1)2 -j , k2 -j ]) = A nk for all n, k.

Recall that f g d = F loc , with F (E) = lim sup n→∞ x n V n (E) where x n = 1/(n log 2) and V n (E) = log #{A nk , A nk ∈ E}.

6.2.1 Prescription on R Theorem 6.1. Let g : R → [0, 1] ∪ {-∞} be an upper semi-continuous function.

There exists a sequence of real numbers (A jk ) j∈N;k∈1..2 j such that g is the large deviation multifractal spectrum of the (A jk ) j,k .

Proof. Let (I n ) n be a sequence of non-empty intervals such that |I n | tends to 0 and R = ∪ n≥N I n for all N . Let i n be a sequence such that for all n , i n ∈ I n (one may for instance choose the midpoint of I n ).

For all n ∈ N, dene A nk , k ∈ [1..2 n ], by: If M (g, I n ) ∈ [0, 1] : A nk = i n , k = 1 . . . 2 n.M (g,In) , A nk = 1 + max(∪ p≤n I p ) , k = 1 + 2 n.M (g,In) . . . 2 n . If M (g, I n ) = -∞ :

A nk = 1 + max(∪ p≤n I p ) , k = 1 • • • 2 n .

We shall apply lemma 6.1 to obtain the result.

For all n, we have: #{A nk , A nk ∈ I n } = 2 n.M (g,In) with the convention 2 -∞ = 0. The map x → log x -log x is bounded on [1, +∞). As a consequence, there exists c such that, for all n ∈ N such that M (g, I n ) = -∞:

V n (I n ) -log(2 M (g,In) ) ≤ c. This entails that, for all n such that M (g, I n ) = -∞:

|x n V n (I n ) -M (g, I n )| ≤ cx n .
Furthermore, when M (g, I n ) = -∞, then V n (I n ) = M (g, I n ).

Thus, d R (x n V n (I n ), M (g, I n ))) tends to 0.

Besides, it is easy to see that, for all n 0 and for all n > n 0 : #{A nk , A nk ∈ I n \ I n 0 } = 0. As a consequence, lim sup n→∞ x n V n (I n 0 \ I n ) = -∞. The assumptions of lemma 6.1 are in force, and the result follows.

  Proposition 4.1 allows to conclude.It is straightforward to check that Proposition 4.3 applies in the paradigmatic cases of multinomial measures ([EM92, BMP92]) and fractal interpolation functions[START_REF] Tricot | The 2-microlocal formalism. Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot[END_REF], yielding the well-known fact that the weak multifractal formalism indeed holds in these situations.

  The continuous 2-microlocal formalism[START_REF] Tricot | The 2-microlocal formalism. Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot[END_REF] is based on the characterization of 2-microlocal spaces from wavelet coecients given in[START_REF] Jaard | Pointwise smoothness, 2-microlocalization and wavelet coecients[END_REF].

20

  In order to facilitate the application of the results of Section 3, we shall dene a localized version of the D w (ρ, ): taking = 0 one gets d(ρ) = {(a, b) ∈ (0, 1) 2 /|b| ρ = max(a, |b|)}.

  Proposition 5.1. ∀s ∈ R, ∀ρ ∈ [0, 1], ∀(a, b) ∈ D w (ρ, ) ∩ (0, 1] × [-1, 1] : 2 -|s | a |s |+ρs ≤ (a + |b|) s ≤ 2 |s | a -|s |+ρs .

Fix > 0 .

 0 There exists n 0 such that I n 0 ⊂ [α -, α + ]. We divide the sequence (I n ) n into two subsequences I σ 1 and I σ 2 dened by:{I σ 1 (n) : n ∈ N} = {I n : n ∈ N, I n 0 ∩ I n = ∅}, {I σ 2 (n) : n ∈ N} = {I n : n ∈ N, I n 0 ∩ I n = ∅}.

)

  Proposition 3.1. Assume that, for a given real number k and for all t ∈ L, U t is k-stable. Let F U be dened by (8). Then F U is stable.Proof. By assumption, for all t ∈ L and all (E 1 , E 2 ) in P(R) 2 ,

Remark Conversely, Lemma 2.2 implies that the multifractal spectrum of any sequence (A jk ) j∈N;k∈1..2 j is an upper semi-continuous function from R to [0, 1] ∪ {-∞}. Thus, we have: Proposition 6.1. The set of all large deviation multifractal spectra coincide with the one of upper semi-continuous functions from R to [0, 1] ∪ {+∞}

Prescription on R

In certain situations, it is convenient to extend the domain of the discrete multifractal spectrum to R. In that view, one sets:

and

Recall the following result from [START_REF] Véhel | On various multifractal spectra[END_REF]:

Theorem 6.1. If g : R → [0, 1] is a large deviation multifractal spectrum, then the supremum of g is 1.

We shall provide a result on the prescription of the large deviation multifractal spectrum in this frame. The proof simplies greatly if we use the following change of variable, that allows to deal smoothly with the particular cases α = ±∞ in the denition of f d g (α): we set, for all θ ∈ [-π/2, π/2],

Dene in an analogous way:

and:

It is then easy to check that, for all θ in [-π/2, π/2]:

with the convention tan(±∞) = ±π/2. We will show that f d g (θ) has maximum 1 on [-π/2, π/2]. Remark rst that, for all θ, all and all j, N (θ, j) ≤ 2 j . As a consequence, 1 is an upper bound to f d g and thus to f d g .

Let us now show that f d g reaches the value 1.

We may divide J 0 = [-π/2, π/2] into two intervals [-π/2, 0] and [0, π/2] such that at least one contains at least 2 j-1 elements in the sequence (arctan A jk ) j,k for an innite number of indices j. Denote this interval J 1 .

Likewise, J 1 may be split into two intervals with diameter not larger than π/4 with at least one interval containing at least 2 j-2 elements in the sequence (arctan A jk ) j,k for an innite number of indices j. Denote J 2 this interval.

Iterating, one gets a sequence of non-empty nested intervals (J m ) m∈N whose diameters tend to 0, which contain at least 2 j-m elements arctan A jk for an innite number of indices j.

Let {θ 0 } denote their intersection, where

contains at least 2 j-m elements arctan A jk for an innite number of indices j. Thus, f d g (θ 0 , ) = 1. This being true for all , we get f d g (θ 0 ) = 1. We conclude that f d g (tan θ 0 ) = 1, with the convention tan(±∞) = ±π/2.

We will now prove the following result: Theorem 6.2. Let g : R → [0, 1] ∪ {-∞} be an upper semi-continuous function dened on R 1 with maximum 1. Then there exists a sequence of real numbers (A jk ) j∈N;k∈1..2 j such that g is the multifractal spectrum of the A jk .

We use the notations (32), (33) and (34) of the preceding part. Let us start by proving a preliminary result: Lemma 6.2. Let g : [-π/2; π/2] → [0, 1] ∪ {-∞} be an upper semi-continuous function on [-π/2, π/2] with maximum 1. Then there exists a sequence of real numbers (A jk ) j∈N;k∈1..2 j such that f d g = g.

Proof. It is straightforward to construct a sequence of intervals (I n ) n such that |I n | tends to 0 and [-π/2, π/2] = ∪ n≥N I n for all N . We also construct a sequence of real numbers i n verifying i n ∈ I n and i n = ±π/2. Let θ max be a point in [-π/2, π/2] such that g(θ max ) = 1.

We build a sequence of points (j n ) n in (-π/2, π/2) such that for all n and all p ≤ n:

We now dene the A nk as:

1 i.e. g is upper semi-continuous on R and it veries in addition g(+∞) ≥ lim sup x→+∞ g(x), g(-∞) ≥ lim sup x→-∞ g(x).

28 For all n: #{A nk , arctan A nk ∈ I n } = 2 nM (g,In) with the convention 2 -∞ = 0. As above, we get that: d R (x n V n (I n ), M (g, I n )) tends to 0 when n tends to innity.

For all m such that θ max / ∈ I m and for all n such that n > m:

and thus lim sup

To obtain theorem 6.2, it now suces to apply lemma 6.2 to g(θ) = g(tan(α)) with the convention tan(±∞) = ±π/2. Remark Conversely, lemma 2.2 yields that for any sequence of real numbers (A jk ) j∈N;k∈1..2 j , the function f d g is upper semi-continuous. In other words, the multifractal spectrum of any sequence A jk is an upper semi-continuous function on R. Furthermore, the multifractal spectrum of any sequence A jk has to reach 1 on R by theorem 6.1

To sum up, we have: Proposition 6.2. A function g : R → [0, 1] ∪ {-∞} is a large deviation multifractal spectrum if and only if it is upper semi-continuous on R and it reaches 1 on R.

2-microlocal Spectrum

The prescription of the 2-microlocal frontier has been considered in [START_REF] Tricot | The 2-microlocal formalism. Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot[END_REF][START_REF] Meyer | Wavelets, vibrations and scaling[END_REF].

Then there exists a distribution f = j,k d jk ψ jk such that g is the 2-microlocal spectrum of f at x 0 .

Proof. We may assume without loss of generality that x 0 = 0. Let (I n ) n be a sequence of open intervals such that |I n | tends to 0 when n tends to innity and [0, 1] ⊂ ∪ n≥N I n for all N . Assume in addition that for all n:

(this condition ensures that each t n is chosen only once).

• t n ∞ < 1/n (this condition ensures that t n tends to 0 when n tends to innity).

To simplify notations, we let: t n = (2 -j(n) , k(n).2 -j(n) ). Note that j(n) and k(n) are integers since we assumed that t n ∈ L d .

Dene nally d jk by: ∀n ∈ N :

From the denition of the d jk , it is easy to check that the 2-microlocal spectrum at 0 of f = j,k d jk ψ jk reads -χ = F loc , where F (E) = lim sup n→∞ (x n V n (E)),

x n = 1/(j(n) log 2) tends to 0 and

Furthermore, for all (n 0 , n), t n ∈ D In , and thus V n (I n 0 \ I n ) = -∞. The assumptions of Lemma 6.1 are veried. As a consequence, χ = g.

Remark

The previous result may be extended to a function g : [0, 1] → R ∪ {±∞} by letting for instance:

Proof. With the same notations as above:

The sequence (d R (x n V n (I n ), M (g, I ∩ I n )) n still tends to 0. Besides, we have again that V n (I m \ I n ) = -∞. As a consequence, Lemma 6.1 still applies.

Remark Conversely, Lemma 2.2 entails that the 2-microlocal spectrum of any distribution at any point is a lower semi-continuous function from [0, 1] to R ∪ {±∞}.

We thus have the following proposition: Proposition 6.3. A function g : [0, 1] → R ∪ {+∞} is a 2-microlocal spectrum if and only if it is lower semi-continuous.