Ngweme Ngakiama

Nkonde Nkiama Joël

Muyer Ayol

Kiyombo Mbela

Etude sur la perception et attitude de la consommation des fruits dans les ménages de la Ville Province de Kinshasa, RD Congo

Keywords: Fruits, acheteurs, vendeurs, marchés Fruits, buyers, sellers, markets

Cette étude a été menée auprès des 124 acheteurs/consommateurs et 126 vendeurs fréquentant les 5 grands marchés de Kinshasa dont le Grand Marché de Kinshasa, Marché de Matete, Marché de Liberté, Marché Gambela et Marché de Selembao choisis parmi tant d'autres marchés de la ville pour une enquête quantitative de type CAP structurée d'un échantillon de 250 enquêtés afin de savoir si leurs ménages consomment ou associent les fruits dans leur alimentation quotidienne. Malgré la disponibilité et la diversité des fruits tropicaux et subtropicaux inondant les marchés locaux de Kinshasa, il s'avère seulement que 18,0 % des ménages des vendeurs et acheteurs enquêtés associent les fruits dans leur alimentation quotidienne.

Introduction

Les réflexions sur la faible consommation de fruits ont longtemps animées les écrits ainsi que les diverses contributions en vue d'améliorer les conditions nutritionnelles de la population. La consommation alimentaire ne peut s'interpréter indépendamment de l'acte alimentaire, acte que l'homme réalise plusieurs fois par jour sous forme de prises ordonnées ou non ordonnées des repas. C'est au cours de cet acte que chaque personne assure à son organisme la couverture de ses besoins, en énergie, en matériaux indispensables, en macro et micro nutriments [START_REF] Friedman | Déterminants de la consommation et des habitudes alimentaires, 1994 2[END_REF].

Selon une étude réalisée en 2007 par le Fonds Mondial de Recherche sur le Cancer et l'Institut Américain de Recherche sur le Cancer dont l'ouvrage nommé « Alimentation, nutrition, activité physique et prévention du cancer : une perspective globale aux États-Unis » visait à étudier les effets des régimes alimentaires auprès des consommateurs américains.

Les techniques de la prévention du cancer, précisent qu'accroitre la production et la consommation de fruits devient une priorité à l'échelle mondiale car sa faible consommation figure parmi les dix principaux facteurs de risque de la mortalité [2]. Cette étude affirme dans ses résultats que sur 100% de cas de maladies enregistrées aux USA pendant les dernières années, ceux causées par la carence en consommation de fruits représentaient 48%.

De même que Hung HC et al [START_REF] Hung | fruit and vegetable intake and risk of major chronic disease[END_REF] dans une étude sur la consommation de fruits et légumes et risque de maladies chroniques majeures estiment que la consommation insuffisante de fruits dans le monde est responsable de 19% de cancers gastro-intestinaux, 31% des cardiopathies et 11% des accidents vasculaires cérébraux.

Quant à Hall et al [START_REF] Hall | Global variability in fruit and vegetable consumption[END_REF] L'accroissement de la consommation des fruits et légumes reste une préoccupation pour les décideurs publics de la santé dans la plupart des pays. Malgré des campagnes qui visent à favoriser la consommation des fruits et légumes, on observe dans de nombreuses études une faible variation de la consommation dans la population. Selon de nombreuses études, un régime riche en fruits et en légumes contribue à la prévention de la maladie cardiovasculaire et de certaines formes de cancer qui comptent parmi les causes principales de décès au Canada [START_REF] Steinmetz | Vegetables, fruit, and cancer prevention: a review[END_REF][START_REF] Gaziano | A prospective study of consumption of carotenoids in fruits and vegetables and decreased cardiovascular mortality in the elderly[END_REF]. Malgré le rôle important de la nutrition dans le maintien d'un bon état de santé, fort peu de données sur les habitudes de consommation des divers aliments dont les fruits ont été recueillies à l'échelle nationale depuis le début des années 1970 [START_REF] Gray-Donald | Food habits of Canadians: Reduction in fat intake over a generation[END_REF][START_REF] Jacobs-Starkey | Food habits of Canadians: comparison of intakes in adults and adolescents to Canada.s Food Guide to Healthy Eating[END_REF].

En République Démocratique du Congo, quoiqu'il y ait existence de nombreux types des fruits cultivés et vendus dans les marchés locaux et dans les alimentations (fruits industrialisés), la population congolaise fait abstraction aux fruits dans les régimes alimentaires quotidiens. Il convient de rappeler l'extrême faiblesse de la consommation congolaise actuelle qui permet de penser que les besoins réels en fruits sont loin d'être satisfaits. Et rares sont les études qui se sont intéressées sur la consommation de fruits dans les ménages [10]. L'objectif de cette étude est de savoir si la population de la ville province de Kinshasa consomme ou associe souvent les fruits dans leur alimentation dans leurs ménages. Le choix de ces marchés était motivé non seulement pour des raisons d'accessibilité, mais ce sont des véritables lieux de brassage de toute la population kinoise. Ils accueillent journellement de milliers des vendeurs et acheteurs qui, dans leurs ménages, sont des consommateurs de toutes sortes des fruits tropicaux et subtropicaux tels que les avocats, les bananes, les bananes plantains et assimilés, les dattes, les figues, les autres fruits tropicaux et subtropicaux (ananas, mangues, goyaves, papayes, fruits de la passion, etc.) ainsi que les olives de table, les noix de coco et les autres fruits oléagineux (coprah, noix et amandes de palmiste… et autres fruits oléagineux) achetés par les ménages pour la consommation.

Matériels et

Sur chaque marché, 50 interviews ont été réalisées dont 25 pour les vendeurs et 25 pour les acheteurs. Ceci nous a permis de comprendre les perceptions et les attitudes de ces 2 groupes ciblés à l'égard de la consommation des fruits. Tous les interviewés ayant participé à l'étude ont donné leur consentement libre et éclairé. La Figure 1 présente la localisation des marchés enquêtés. Toutes les analyses ont été effectuées en considérant un niveau de signification de 5%.

Les données collectées ont porté sur les perceptions et les attitudes des ménages (vendeurs et acheteurs) quant à la consommation des fruits dans les ménages tels que la nature des fruits, la consommation des fruits, la fréquence des achats des fruits, types de fruits et le lavage des fruits avant la consommation.

Résultats

Caractéristiques sociodémographiques

La majorité des enquêtés (vendeurs et acheteurs/consommateurs) était de sexe féminin avec un pourcentage de 88,4. Leur âge moyen était de 37,33 ans, un écart-type de 10,907 ans avec les extrêmes de 18 et 68 ans. Les vendeurs (17,2%) étaient âgés de 41 à 51 ans. Les tranches d'âge de 30 à 40 ans (14,0%) et 52 ans et plus représentaient 12,4 %. En ce qui concerne les acheteurs, 22,0 % étaient âgés de moins de 30 ans et 20,0 % de 30 à 40 ans.

Quant aux autres aspects sociodémographiques, les résultats de cette étude ont montré que la majorité des vendeurs (34,0%) avaient achevé le niveau secondaire et 5,6 % le niveau universitaire ; 34,4% des acheteurs avaient achevé le niveau secondaire et 9,6 % le niveau universitaire. En ce qui concerne le statut matrimonial, l'étude a montré que 30,8% des vendeurs étaient mariés et 8,4% des célibataires. Pour les acheteurs, 17,6 % étaient des mariés et 26,4% des célibataires.

Consommation des fruits dans les ménages

La consommation des fruits dans les ménages des vendeurs et acheteurs est liée à leurs habitudes alimentaires. Pour ceux qui consomment très souvent les fruits, le taux est faible à 18,0 %, 46,0 % sont ceux qui les consomment souvent et 17,3 % sont les ménages qui les consomment occasionnellement. Il est à signaler qu'une catégorie des ménages qui consomment les fruits rarement (18,4%) (Figure 1).

Types des fruits consommés dans les ménages

Cette étude montre que 86,0 % des ménages des vendeurs et acheteurs consomment en premier rang les mangues, 67,6 % des ménages, les oranges, 50,8 % les avocats et 45,2 des ménages consomment les bananes (Figure 2).

Connaissances sur la nature des fruits consommés

A la question de savoir sur la qualité des fruits consommés dans les ménages des vendeurs et acheteurs, 58,8 % des interviewés disent que les fruits sont naturels et 1,2 % les classent dans la catégorie des produits OGM (Figure 3).

Fréquence de consommation des fruits

Le taux de fréquence de la consommation des fruits dans les ménages des vendeurs et acheteurs est de 25,2 % pour tous les jours, 33,2 % pour deux fois par semaine et un taux de consommation de 9,6 % une fois par semaine (Figure 4).

Attitudes des vendeurs et acheteurs sur le lavage des fruits dans les ménages

La majorité des ménages des vendeurs et acheteurs enquêtés ont l'habitude de laver les fruits avant la consommation, 36,4 % les font parfois et 12,0 % des ménages ne les font pas (Figure 6).

Figure 1 :

 1 Figure 1 : Carte de localisation de la République démocratique du Congo, les zones d'étude des différents marchés visités

Figure 1 :

 1 Figure 1 : Consommation des fruits dans les ménages des acheteurs et acheteurs

Figure 2 :

 2 Figure 2 : Types des fruits consommés par les ménages des vendeurs et acheteurs

Figure 3 :

 3 Figure 3 : Types sur la nature des fruits consommés par les ménages des vendeurs et acheteurs

Figure 4 :

 4 Figure 4 : Fréquence de consommation des fruits dans les ménages des vendeurs et acheteurs

Figure 5 :

 5 Figure 5 : Endroits pour les achats des fruits

Figure 6 :

 6 Figure 6 : Lavage des fruits avant la consommation


Cl Ément Zrounba
email: clement.zrounba@ec-lyon.fr

Raphael Cardoso

Maurício Gomes De Queiroz

Paul Jimenez

Mohab Abdalla

Alberto Bosio

Ébastien Le Beux

Fabio Pavanello

Ian O'connor

Introducing SPECS: Scalable Photonic Event-driven Circuit Simulator

We present SPECS, a SystemC-based tool for fast, fs-accurate large-scale photonic circuit simulation. Through its event-driven nature, SPECS excels in sparse activity applications like neuromorphic computing. Even in unfavorable scenarios, it is faster than other simulators by orders of magnitude. ©2023 The

Introduction

As photonic technology takes on an increasingly significant role in various applications in telecommunications, sensing, and computing [START_REF] Jalali | Silicon Photonics[END_REF], [START_REF] Thomson | Roadmap on silicon photonics[END_REF] , accurate and efficient simulation tools become essential for designing, verifying, and optimizing photonic components and systems [START_REF] Chrostowski | Silicon Photonics Design[END_REF] .

Photonic simulations rely on a multi-level abstraction methodology similar to the one prevalent in electronics [START_REF] Khan | Photonic Integrated Circuit Design in a Foundry+Fabless Ecosystem[END_REF] , and in fact re-purposes many tools from the electronic design automation (EDA) portfolio [START_REF] Ye | SPICE-Compatible Equivalent Circuit Models for Accurate Time-Domain Simulations of Passive Photonic Integrated Circuits[END_REF]- [START_REF] Chrostowski | Silicon Photonic Circuit Design Using Rapid Prototyping Foundry Process Design Kits[END_REF] . At the device level, 2D/3D electromagnetic simulations provide detailed information about the behavior of individual components. This information, crossed with device measurements, is used to create simplified device models for use in circuit simulators such as Lumerical TM Interconnect [8] , Luceda Caphe (now part of the IPKISS framework) [9] , or Photontorch [START_REF] Laporte | Highly parallel simulation and optimization of photonic circuits in time and frequency domain based on the deep-learning framework pytorch[END_REF] . Finally, system designers can simplify circuit behavior into system-level metrics (e.g., bandwidth, latency...) to perform very large-scale analyses [START_REF] Sepulveda | Communication Aware Design Method for Optical Network-on-Chip[END_REF] . These methods have been proven useful, but result in prohibitive computational costs (at circuit level) or modelling efforts (at system level) when designing large-scale photonic circuits, calling for an intermediate abstraction level between circuit and system, still inexistent in photonics [START_REF] Ferreira De Lima | Primer on silicon neuromorphic photonic processors: Architecture and compiler[END_REF] .

In electronics, industry standards for hardware description languages (HDL) [START_REF] Ashenden | The designer's guide to VHDL[END_REF] were instrumental to enabling the design of large systems. These standards were further improved with analog and mixed-signal (AMS) extensions to provide both continuous-time and event-driven modeling primitives in a single package [START_REF] Christen | Vhdl-ams-a hardware description language for analog and mixed-signal applications[END_REF] . In event-driven solvers, components are simulated only when necessary, significantly reducing computational costs and enabling more efficient analysis of large-scale systems, without degrading temporal accuracy. This approach seems particularly well-suited for optical systems, which often exhibit temporally and/or spatially sparse activity [START_REF] Shastri | Photonics for artificial intelligence and neuromorphic computing[END_REF] , owing to the scale difference between the speed of signals and the bandwidth of supporting electronics. While there are priors for event-driven simulation of photonic circuits [START_REF] Rumley | PhoenixSim: Crosslayer Design and Modeling of Silicon Photonic Interconnects[END_REF]- [START_REF] Briere | Heterogeneous Modelling of an Optical Network-on-Chip with SystemC[END_REF] , they focus on the interconnect layer and abstracted systemlevel metrics.

In this contribution, we introduce SPECS (Scalable Photonic Event-driven Circuit Simulator) [19] , an open-source simulation tool that leverages the event-driven paradigm to overcome performance limitations of traditional time-driven simulation methods. We first present the rationale between our approach and brief implementation details of SPECS. We then demonstrate the effectiveness of SPECS by studying a Coupled-Resonator Optical Waveguide (CROW) circuit and compare it with Interconnect and Photontorch (PT).

Proposed Approach

SPECS is built upon the SystemC framework, a widely adopted hardware description language (HDL) and simulation library for system-level design [START_REF]SystemC standard[END_REF] . By utilizing SystemC, we facilitate the integration of SPECS with other system-level design tools and enable efficient co-simulation of photonic and electronic subsystems.

For circuit description, SPECS employs SPICElike netlists, standard in electronic circuit simulators. This choice ensures ease of integration with a broad range of existing EDA tools and allows designers to leverage their knowledge and experience. Furthermore, this approach enables hier- archical descriptions of photonic circuits, simplifying the design process and facilitating the cosimulation of complex systems. While the current implementation of SPECS focuses on event-driven simulation, the framework can be extended to support mixed-mode simulations for cases where non-linear effects impose a more detailed analysis or, inversely, used to extend existing time-driven frameworks. Mixedmode simulations would enable a more comprehensive coverage of various photonic circuit scenarios while still benefiting from the efficiency advantages of event-driven simulations.

A typical use of SPECS is shown in Figure 1. Existing tools, here KiCAD [START_REF] Kicad | [END_REF] , can be adapted to edit circuit schematics and generate a SPECS netlist, which can be used to perform frequencyor time-domain simulations.

Implementation Details

SPECS, like other photonic circuit simulators, adopts a "baseband" approach and represents optical signals by a complex field amplitude E(t) and a carrier wavelength λ 0 . Waveforms are reduced to a sequence of square pulses which in turn are decomposed into at most two "events", as shown in Figure 2a. It should be noted that event-driven simulation is naturally more suited to digital waveforms than continuously varying ones. When a circuit component receives an event, it calculates the corresponding outputs and schedules their emission based on component-specific delays as illustrated in Figure 2b.

SPECS defines many basic components, such as waveguides, couplers, phase-shifters, generic S-matrix devices, as well as arbitrary sources and probes. The implementation of these components is straightforward and functionally equivalent to other circuit simulators [START_REF] Laporte | Highly parallel simulation and optimization of photonic circuits in time and frequency domain based on the deep-learning framework pytorch[END_REF] . New compo- nents can be added by modifying the C++ source code or using a generic Python module for fast prototyping.

For purely feedforward circuits with digital inputs, SPECS approaches the optimal simulation time by design. For circuits involving feedback, it can still perform very well, provided that the balance between accuracy and runtime is chosen wisely. Several cursors are available for a user to tune the accuracy of simulations: • Temporal resolution -sets the precision with which events are resolved. Decreasing resolution can greatly reduce simulation time, but will lower the precision of the final waveforms by masking what happens between events. • Absolute and Relative tolerance -relax precision constraints by specifying a threshold below which small changes do not need to be emitted by devices. Without them, events representing infinitesimal changes in field values could propagate indefinitely in circuits with feedback. These should not be conflated with similarly named settings of time-driven simulators which, even though they share similar goals, typically control the time-step of the solver. Lastly, both frequency-domain simulation and impulse-response calculation constitute special use-cases of SPECS and can be performed easily and efficiently without external software.

Results and Discussion

To evaluate the performance of SPECS, we studied a CROW circuit, shown in Figure 3a, made up of N chained ring resonators. This circuit is composed of one source emitting 1 mW, 1 ns pulses at 500 MHz, 2N waveguides with lengths L i = (33.2 ± 5.0) µm sampled from a normal distribution, and N + 1 directional couplers with κ = 0.15 power cross-coupling ratio. Simulations were run on a single thread using a modern laptop powered by an Intel i7-1165G7 2.8 GHz CPU and 16 GB of RAM. Interconnect, PT and SPECS were all used as-is, with no platformspecific or circuit-specific tuning. SPECS' absolute and relative tolerances were respectively set to 10 -8 V m -1 (0.1 fW) and 0.01 %.

We first verify our benchmarking environment by comparing the through response for N = 3 and very long delays (L ≈ 350 µm). Results, presented in Fig. 3, show that Interconnect surprisingly did not match the response yielded by both PT and SPECS at 100 fs resolution. At 1 fs resolution, all simulations yielded nearly identical results, with pair-wise root-mean-square errors below ∼ 3 µW, although SPECS was 6290× (resp. 5090×) faster than PT (resp. Interconnect).

Finally, we turn to CROW circuits with up to N = 512 rings. In PT and Lumerical, we simulate for a duration of 3 ns (30000 timesteps at 100 fs). In SPECS, we instead simulate 10 equally-spaced 1 ns pulses (20 ns) and normalize the results to accurately capture the effect of events continuing to propagate beyond the initial duration and accumulating with subsequent pulses. Execution times for 3000 timesteps (300 ps), averaged over 100 runs for SPECS, are shown in Figure 3c. The whiskers in Figure 3c show the extrema measured for SPECS, due to the random sampling of waveguide lengths, illustrating the circuitdependent variability in simulation times.

Although CROWs constitute an unfavorable scenario for event-driven simulators, SPECS' performance remained order of magnitudes better than PT and Interconnect, in part because beyond N ∼ 20, simulation time saturates around 40 ms due to losses preventing any meaningful signal (according to SPECS' tolerances) to propagate too far in the CROW. This native input dependent optimization is precisely what sets it apart from the state of the art.

We were not able to independently run a GPUaccelerated version of PT, but reported performances for this use-case [START_REF] Laporte | Highly parallel simulation and optimization of photonic circuits in time and frequency domain based on the deep-learning framework pytorch[END_REF] are still an order of magnitude slower than SPECS running on a single thread.

Conclusions

Event-driven photonic simulation can be a powerful tool, and likely the next step for simulating large photonic circuits driven by digital waveforms, or exhibiting sparse activity.

We showed that our implementation outperforms off-the-shelf tools by a significant margin in terms of simulation speed while offering functionally similar accuracy. In particular, its performances scale very well with circuit size due to its naturally frugal simulation approach.

It is important to note that different simulators serve different purposes. For example, SPECS cannot efficiently optimize circuits through backpropagation like Photontorch does. Additionally, mixed-mode simulations will be necessary to cover more complex circuits.

Finally, through SystemC and a versatile netlist interface, SPECS can enable co-simulation of computing cores and photonic accelerators, or be integrated into other simulation platforms.

Power* 6 Fig. 1 :

 61 Fig. 1: Example design and simulation flow: (a) schematic description of the circuit, (b) the corresponding SPECS netlist, (c) frequency-and time-domain simulations of the circuit.

Fig. 2 :

 2 Fig. 2: (a) Digital signals are inefficiently represented in time-driven (top) simulators compared to event-driven (bottom) ones and (b) architecture of a unidirectional 2-port device in SPECS.

Fig. 3 :

 3 Fig. 3: Benchmark results (a) architecture of the CROW circuit used for benchmarking, (b) time-domain results for the same CROW (N = 3) on all simulators, at 100 fs and 1 fs (c) single-threaded CROW simulation times at 100 fs.

Acknowledgments

This work was funded by ANR under grants ANR-18-CE24-0027 (OPTICALL 2 ) and ANR-20-CE24-


Sylvain Durand
email: sylvain.durand@mi.parisdescartes.fr

On the construction of discrete directional wavelets: HDWT, space-frequency localization and redundancy factor

Keywords: Wavelet transform, directional wavelets, two-dimensional filter banks AMS Subject Classifications. 42C40, 65T60, 68U10, 94A08

We address the issue of constructing directional wavelet bases. After considering orthonormal directional wavelets whose Fourier transforms are indicator functions, we give a construction of directional wavelets with fast decay that is based on an hexagonal filter bank tree. An implementation for squarely sampled images and numerical results are presented. Then we discuss the frequency localization of directional wavelet bases. We analyze the incompatibility between a proper localization and the non-redundancy constraint, and show that the non permissibility condition can be extended to general wavelet bases that are not necessary generated by a filter bank tree. At last, we show that there exist directional wavelet tight frames that are well localized and have a redundancy factor arbitrary close to 1.

Introduction

Because of their ability to characterize piecewise smooth functions with few coefficients, wavelets have been a successful tool in several signal and image processing tasks. Wavelets for 2D images are typically constructed by tensor products of 1D wavelets. However the obtained separable wavelets do not provide sparse representations of every image's structures. Indeed a common image model is based on the observation that if one zooms in on a detail of an ordinary image, its geometrical flow is generally, nearly uniform. In other words, its local Fourier transform is concentrated along a line which is orthogonal to the flow (see Fig. 1). When the image is decomposed in a separable wavelet basis, its energy is spread on many coefficients (see Fig. 2-(a)). Moreover, both diagonal directions are ambiguously represented by a single wavelet. Although the wavelet transform provides sparse (but not optimal) representations of horizontal and vertical singularities (using the horizontal and the vertical wavelets), it does not permit to represent efficiently singularities which are positioned along other directions.

For more than a decade, new transforms have been proposed in order to make up for this lack of directionality of separable wavelets. Several adaptative transforms were developed. For instance, first generation bandelets [START_REF] Pennec | Sparse geometrical image representation with bandelets[END_REF] consist in warping separable wavelets along the geometrical flow of images, and performing a bandeletization (see Fig. 2-(b)). We focus in this article on the construction of non adaptative transforms that are generally implemented in a faster algorithm as they do not require prior information on the image. Among the most prominent non-adaptative, directional transforms, one may mention hexagonal wavelets [START_REF] Simoncelli | Subband Image Coding with Hexagonal Quadrature Mirror Filters[END_REF][START_REF] Cohen | Compactly supported bidimensional wavelet bases with hexagonal symmetry[END_REF], directional filter banks [START_REF] Bamberger | A filter bank for the directional decomposition of images: theory and design[END_REF], steerable pyramids [START_REF] Simoncelli | Shiftable Multiscale Transforms[END_REF], directional wavelets [START_REF] Murenzi | Ondelettes multidimensionnelles et applications à l'analyse d'images[END_REF][START_REF] Antoine | Two-dimensional directional wavelets and the scale-angle representation[END_REF], brushlets [START_REF] Meyer | Brushlets: a tool for directional image analysis and image compression[END_REF], dual-tree complex wavelets [START_REF] Kingsbury | The dual-tree complex wavelet transform: A new technique for shift invariance and directional filters[END_REF][START_REF] Selesnick | The dual-tree complex wavelet transform -A coherent framework for multiscale signal and image processing[END_REF], ridgelets [START_REF] Candès | Ridgelets -a key to higher dimensional intermittency ?[END_REF], curvelets [START_REF] Candès | Curvelets -a suprizingly effective nonadaptive representation for objects with edges, Curve and Surface Fitting[END_REF] (see Fig. 2-(c)) and contourlets [START_REF] Do | Contourlets, Beyond Wavelets[END_REF]. All these transforms are of great interest for applications in fields such as compression, denoising, enhancement and oriented feature extraction. Their construction leaves however some open questions. Indeed, in compression, powerful and practical transforms must satisfy the following requirements But none of the quoted above transforms satisfies all these properties. The last one is crucial in the approach we adopted. Curvelets and contourlets are known to perform very well at low bit rate compression. But they are redundant frames. As a consequences, separable wavelets remain sometimes more efficient to compress images at rates that are used in practice. Let us emphasize that, in this article, we are thinking of compression applications. If one deals with denoising, for instance, the constraint "no redundancy" has to be replaced by "shift invariance". Directional wavelets were introduced in [START_REF] Murenzi | Ondelettes multidimensionnelles et applications à l'analyse d'images[END_REF]. A family of wavelets is generated from a single function, called mother wavelet, by translation, dilation and rotation,

a ψ(aR θ • -b), a ∈ R + , b ∈ R 2 , R θ ∈ SO(2, R).
The 3 parameters a, b, θ can be discretized in order to obtain tight frames for L 2 (R 2 ). Curvelets are constructed with a more subtle family of rotation-dilation operators. Indeed it is proved in [START_REF] Candès | Curvelets -a suprizingly effective nonadaptive representation for objects with edges, Curve and Surface Fitting[END_REF] that one obtains sparser representations of some classes of images by using functions that have a finer angular resolution at finer scales, following a parabolic scaling law. (At scale a the angular resolution must be O( √ a).) For that reason, we consider the more general family (that we still call directional wavelets)

{| det D k | ψ(D k • -γ)} k∈K,γ∈Γ = {ψ k ( • -γ)} k∈K,γ∈Γ k
where Γ is a regular lattice on R 2 and, for all k, andD k is a linear operator on R 2 . The sequence {D k } k∈K can be constructed following the theory of wavelets with composite dilations [START_REF] Guo | Wavelets with composite dilations and their MRA properties[END_REF] although more general sequences are considered in this article. In order to complete the basis, one has to consider also a low frequency function φ that we can denote ψ 0 but that it is not obtained by dilating the mother wavelet ψ. At last, in order to obtain bases for L 2 (R 2 ), we will not always assume that

ψ k = | det D k | 1/2 ψ(D k • ), Γ k = D -1 k Γ,
ψ k = | det D k | 1/2 ψ(D k • ) but that ψ k is close to | det D k | 1/2 ψ(D k • ) in
a sense that will be specified below. Notice that the same kind of hypothesis is made to obtain second generation curvelets tight frames [START_REF] Candès | Fast Dicrete Curvelet Transform[END_REF].

In Section 2, the construction of directional wavelet bases is restricted to wavelets that are Fourier transforms of indicator functions. In order to meet the above requirements, these wavelets have to be approximated by functions with fast decay. In that direction, we presents a numerical implementation of one of the transforms introduced in [START_REF] Durand | M-band filtering and non-redundant directional wavelets[END_REF]. It is implemented in a fast algorithm that is based on a filter bank tree with hexagonal and 2-channels filter banks. This hexagonal directional wavelet transform is defined in Section 3.1. In Section 3.2, we propose an implementation for squarely sampled images, while in Section 3.3, we give some numerical results and comparisons. In Section 4, we consider the frequency localization of directional wavelets with fast decay. While it satisfies all the other, cited above, required properties, the proposed hexagonal directional wavelet transform fails to achieve a perfect frequency localization. General wavelets that are not necessary generated by filter bank trees are considered in Section 4.2. The incompatibilities between all the required properties is addressed in a statement that generalizes the permissibility condition for filter banks. In order to circumvent this obstruction in the construction of directional wavelet bases, we show, in Section 5, that there are directional wavelet tight frames that are well localized in the space and in the frequency domains, and that have arbitrary small redundancy factors. At last, a conclusion is drawn in Section 6.

Shannon Directional Wavelet Bases

As it is shown in Figure 2, the issue of obtaining sparse representations of images can be addressed by constructing tilings of the frequency domain. In this section, the frequency localization is therefore singled out as one consider wavelets whose Fourier transforms are indicator functions. The next sections are devoted to the issue of approximating these basis functions with wavelets which are well localized in space and frequency domains.

Assume the Fourier transform of ψ s k is

ψ s k = 2π|A k | -1/2 χ A k where A k is an open subset of R 2 , |A k | is its area and χ A k is the indicator function of A k .
Recall that the dual lattice Γ * of a lattice Γ ⊂ R 2 is defined by

Γ * = {γ ∈ R n : η, γ ∈ 2πZ, ∀η ∈ Γ},
so that the Fourier transform is an isomorphism between ℓ 2 (Γ) and L 2 (R 2 /Γ * ). (See the appendix for more informations on the Fourier transform on regular lattice.)

Proposition 1 The family {ψ s k ( • -γ)} k∈K,γ∈Γ k is an orthonormal basis for L 2 (R 2 ) if and only if 1. {A k + {γ}} γ∈Γ * k is a tiling of R 2 , 2. {A k } k∈K is a tiling of R 2 .
Proof. The main idea of the proof of this proposition is that Condition 1 is satisfied if and only if {ψ s k ( •γ)} γ∈Γ k is an orthonormal basis for and Condition 2 is equivalent to

W k = {f ∈ L 2 (R 2 ) : supp f ⊂ A k }, 0 (a)
L 2 (R 2 ) = k∈K W k ,
where the sum is orthogonal. Details of the proof are left to the reader. ⋄ Except possibly for one set, the sets A k are obtained by dilating a single set A.

(A k = D * k A where D * k is the adjoint operator of D k .
) In order to have real valued wavelets, we assume that A is symmetric about 0, and in order to have a fine angular resolution, A must be included in a cone. An interesting example is given by

A = {ξ ∈ R 2 : |ξ 1 | < α|ξ 2 | and 1 < |ξ 2 | < 2},
for some α > 0. One checks easily that Condition 1 of Proposition 1 is satisfied with Γ * = Ze 1 + Ze 2 where e 1 = (0, 2) and e 2 = (3α, 1) (see Fig. 3-(a)). With good choices for α and K ∈ N, Condition 2 is satisfied by applying to A, dilations of factor 2 (and 1/2) and rotations of angle π/K (see Fig. 3-(b)). Thus, if ψ s = 2π|A| -1/2 χ A and R is the rotation of angle π/K, then

{2 j ψ s (2 j R k • -γ)} j∈Z,k∈{1,•••,K},γ∈Γ
is an orthonormal basis for L 2 (R 2 ). There are many other examples that satisfy the conditions of Proposition 1. The same set A can be dilated in such way that one obtains the tiling of Fig. 2-(c), generating what could be called a Shannon curvelet basis. Notice that, more generally, when going from one scale to a finer one, the number of directions can be multiplied by any integer number. There are also other examples of sets A defining mother wavelets ψ s . One has, for instance, p/(p -1)-adic directional wavelets, with p ∈ N, by using

A = {ξ ∈ R 2 : |ξ 1 | < α|ξ 2 | and p -1 < |ξ 2 | < p}
that forms a tiling of R 2 when it is shifted along Γ * = Ze 1 + Ze 2 where e 1 = (0, 2) and e 2 = ((2p -1)α, 1)

Regular Directional Wavelet Bases

We address below the construction of regular wavelets (in the sense of wavelets with fast decay) {ψ k } k∈K that approximate Shannon wavelets {ψ s k } k∈K . The term approximate is used here in the sense that the loss of energy is controlled :

ψ k -ψ s k 2 2 < ǫ, for ǫ small. (1)
Necessary and sufficient conditions for a family {ψ k ( •γ)} k∈K,γ∈Γ k , with general

ψ k ∈ L 2 (R 2 )
, to be a basis for L 2 (R 2 ) can be found (see Prop. 4 below). The difficulty is however to find wavelets ψ k that satisfy these conditions. An easy way to construct directional regular wavelets is to consider wavelets implemented in filter bank trees, as their design is reduced to the construction of filter banks. This idea was developed in [START_REF] Durand | M-band filtering and non-redundant directional wavelets[END_REF] and several examples of regular directional wavelet basis were given. Close ideas were also proposed in [START_REF] Nguyen | Multiresolution Direction Filter Banks: Theory, Design and Applications[END_REF]. We develop, in this section, one of the examples introduced in [START_REF] Durand | M-band filtering and non-redundant directional wavelets[END_REF]. Numerical results and comparisons are given.

Hexagonal Directional Wavelets

Hexagonal wavelets were introduced in [START_REF] Simoncelli | Subband Image Coding with Hexagonal Quadrature Mirror Filters[END_REF] and [START_REF] Cohen | Compactly supported bidimensional wavelet bases with hexagonal symmetry[END_REF]. These wavelets are obtained one from another by dilation, translation and rotation of ∓2π/3, and are therefore an example of directional wavelet basis in the sense of [START_REF] Antoine | Two-dimensional directional wavelets and the scale-angle representation[END_REF] with 3 directions. They are designed for images that are sampled on the regular hexagonal lattice

Γ = 1 0 1/ √ 3 2/ √ 3 Z 2 .
Let us recall that the hexagonal reciprocal lattice is

Γ * = 2πM Z 2 where M = 1 -1/2 0 √ 3/2 . ( 2
)
and that the associated reciprocal cell has the shape of a regular hexagon. The space L 2 (R 2 ) is decomposed into multiscale and multidirectional subspaces

L 2 (R) = j∈Z k∈{1,2,3} W k j = j∈Z k∈{1,2,3} d j R k W,
where d j is the dilation of factor 2 j and R is the rotation of angle 2π/3. Each subspace is spanned by a mother wavelet ψ k that is shifted along an hexagonal lattice at a given scale. More precisely {2 j ψ k (2 j • -γ) : γ ∈ Γ} is a Riesz basis or an orthonormal basis for W k j . These wavelets are generated by a multiresolution analysis (MRA) [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF]

(V j ) j∈J for L 2 (R 2 ) with a scaling function φ such that {2 j φ(2 j • -λ) : λ ∈ Γ} is a basis for V j . The decomposition V j = V j-1 ⊕ W 1 j-1 ⊕ W 2 j-1 ⊕ W 3 j-1
leads to the equations for almost every ω ∈ R 2 , where

φ(ω) = 1 2 H 0 (ω/2) φ(ω/2), ψ k (ω) = 1 2 H k (ω/2) φ(ω/2), ∀k ∈ {1, 2, 3},
H k ∈ L 2 (R 2 /Γ * )
, for all k ∈ {0, . . . , 3}. In order to have an orthonormal basis, the conjugate mirror filters (H k ) k∈{0,1,2,3,} should be designed in such a way that the matrix

(H k (ω + γ)) k∈{0,1,2,3},γ∈(0.5Γ * )/Γ * is unitary for almost every ω ∈ R 2 , where (0.5Γ * )/Γ * = {(0, 0), (π, 0), (π/2, √ 3π/2), (-π/2, √ 3π/2)}.
In the case of a discrete image sampled on Γ, wavelet coefficients are computed via a filter bank tree. The above conjugate mirror filters define a filter bank whose frequency selectivity is symbolized in Figure 4-(a). It decomposes the original image onto 4 subbands. At each level, the same filter bank is applied again to the low frequency subband only. One obtains this way a transform corresponding to the frequency partitioning of Figure 4-(b).

The Hexagonal Directional Wavelet Transform (HDWT) that we propose consists in dividing each directional fluctuation space W k j in more subspaces with finer directions

W k j = l∈{1,•••,2 p } W k,l j .
This operation is achieved by applying specific filter banks to the high frequency channels of Figure 4-(b). Indeed if we apply the filter bank of Figure 5-(a) to the high frequency subband ♯1, the latter can be divided into 2 new directional subbands. We can clearly apply the same process to the 2 other high frequency subbands in order to obtain 6 directions as in Figure 6-(a). Notice that the filters used at this step can be designed by dilating and shifting the well known diamond shaped quincunx filter.

Each of these 6 subbands can be divided again into 2 other subbands in order to obtain 12 directions. Apply for example the filter bank of Figure 5-(b) to the subband ♯2 of Figure 6-(a). By iterating the process, we can have 3×2 p directions where p ∈ N. A parabolic scaling law [START_REF] Candès | Curvelets -a suprizingly effective nonadaptive representation for objects with edges, Curve and Surface Fitting[END_REF] can be applied to the transform as in Figure 6-(b).

1 0 0 2 1 0 2 0 ( ) ( ) 0 0 0 0 ( ) ( ) 1 2 1 2 (a) (b)
Figure 5: Two filter banks used in the proposed transform.

Implementation on square lattice

The question of filter's design is not developed in this article. Several methods have already been proposed in order to construct non-separable filter banks. Most of them are restricted to 2 channels. For example, generalized McClellan transform [START_REF] Tay | Flexible design of multidimensional perfect reconstruction FIR 2-band filters using transformations of variables[END_REF] permits to design transfer functions whose essential support approximate some arbitrary set. In [START_REF] Cohen | Compactly supported bidimensional wavelet bases with hexagonal symmetry[END_REF], a 4-channel biorthogonal, hexagonal filter bank was constructed. More recently, a method has been proposed to obtain 2-channel filter banks with directional vanishing moments [START_REF] Cunha | On the two-channel filter banks with directional vanishing moments[END_REF]. However the design of M-band filters with finite impulse responses and an arbitrarily small angle between two consecutive directional vanishing moments remains an open question.

As our objective is to construct wavelets that approximate the associated Shannon wavelets, we do not use the hexagonal filter of [START_REF] Cohen | Compactly supported bidimensional wavelet bases with hexagonal symmetry[END_REF] that is not sufficiently well localized in the frequency domain. In the experiments below, all the filters involved in the transform have been designed using the equation

H 0 = | det D| 1/2 g * χ A0 γ∈(D -1 Λ * )/Λ * |g * χ A0 ( . + γ)| 2 1/2 , (3)
where A 0 is the ideal support of H 0 , D is the dilation matrix of the filter bank and Λ is the lattice that supports the signal to be filtered. We chose for g a Gaussian function.

It is approximated by a compactly supported function. Such a design method is a simple way to obtain simultaneously orthonormality, space and frequency localization, linear phase and nearly vanishing directional moments. Moreover, when g tends to a Dirac measure δ 0 , H 0 tend to the associated Shannon filters in the sense that

H 0 -| det D| 1/2 χ A0 2 → 0.
In other words, Condition (1) can be satisfied for arbitrarily small ǫ. These filters have however infinite impulse responses. They are therefore implemented in the Fourier domain. It may seems to be a drawback of the proposed implementations. There are however several reasons for using the Fourier transform. First, in order to have a fine directionality one should use filters that are very well localized in the frequency domain and that have therefore large space support. If the associated filter bank is implemented in the spatial domain, the computation time of the algorithm is proportional to the length of that support and might be larger than the computation time of a FFT (although it is O(N ) where N is the size of the image). Secondly, if O(log N ) directions are set at the finest scale, the computation time of the whole transform that is implemented in the spatial domain is actually O(N log N ). A major argument for using the Fourier transform is that images are in general sampled on a square lattice. The Fourier transform is a natural tool to re-sample images on an hexagonal lattice as it comes down to shifting their spectrum. This point is developed further below. At last recall that the curvelets are also bandlimited and are implemented in the Fourier domain. In some sense, the transform remains faithful to the curvelets although it generates some aliasing problems that are addressed later. By using bandlimited wavelets, we can measure in some way the magnitude of this aliasing.

As it is alluded to above, the algorithm is implemented for images that are sampled on the square lattice Z 2 . There are at least two ways to implement hexagonal directional wavelets on a square lattice. A first method consists in changing the filters H k into H k • M where M is given in [START_REF] Bamberger | A filter bank for the directional decomposition of images: theory and design[END_REF]. If one assumes that the spectrum of images is supported on the circumscribed disk of the square reciprocal cell, then there is nearly no aliasing in the obtained transform as it is shown in Figure 7-(b). The transform can moreover be implemented with finite impulse response filters. However the obtained refinement filter is clearly not enough isotropic.

In the experiments below, we follow another approach. A standard Fourier transform is applied to the image. The obtained transform is however considered as the Fourier transform of an image sampled on the irregular hexagonal lattice

Γ = 1 0 1/2 1 Z 2 .
It amounts to exchanging the periodization vectors (2π, 0) and (0, 2π) for (2π, 0) and (π, 2π). The Fourier coefficients remain the same, they are just shifted. As it is shown in Figure 7-(c), one can keep the same reciprocal cell. Notice that the image could also be re-sampled on a regular hexagonal lattice without deteriorating the circular spectrum of the image. The number of samples to represent the image would be reduced that way by a factor √ 3/2. This idea is not developed in the article as our objective is to design bases for ℓ 2 (Z 2 ). The reciprocal cells of Figure 7 have the same surface. Thus there is no lost of data when exchanging the lattices whatever the shape of the frequency support of the image. Moreover this process transforms an orthonormal basis into an orthonormal basis.

All the filter banks are implemented in the Fourier domain. Let X be the Fourier transform of an input, discrete 2D signal x. For each filter bank, the output y k of the k th channel is given by its Fourier transform

Y k = γ∈(D -1 Λ * )/Λ * H k ( • + γ) X( • + γ).
At the end of the process, an inverse Fourier transform is applied to each subband. A different Fourier transform is used for each lattice. They all run in an O(N log N ) algorithm for an image of size N (see [START_REF] Mersereau | The processing of hexagonally sampled two-dimensional signal[END_REF] for an hexagonal FFT, and the appendix for a FFT on a general, regular lattice).

The hexagonal filter is designed using (3) with One checks easily that the associated matrix is unitary. The 2-channel filter banks are designed by the same method. One set

A 0 = {ω ∈ R 2 : |ω 1 | < π/2 & |2ω 1 + 3ω 2 | < 2π & |2ω 1 -3ω 2 | < 2π} + Γ * ,
H 1 (ω) = e i ω,δ H 0 (ω + γ),
where {γ} = (D -1 Λ * )/Λ * and δ is such that γ, δ = π modulo 2π. The hexagonal directional filter banks have been implemented up to 12 directions. Figure 8 shows the Barbara image that have decomposed onto 12 directional subbands and a low frequency subband.

Numerical results

We present in this section some numerical results obtained with the HDWT, and comparisons with the Hexagonal Wavelet Transform (HWT) and the 9/7 Separable Wavelet Transform (SWT) [START_REF] Cohen | Biorthogonal bases of compactly supported wavelets[END_REF]. Let us point out that the compression performance of the 9/7 tap filters are widely recognized and that it has been incorporated in the JPEG 2000 standard. On the other hand, we do not show the best possible implementation The HWT consists in choosing 3 directions at each level of the HDWT.

In Figure 9 we show non linear approximations of the 512 × 512 Barbara image. For the 3 transforms, the 10 4 largest coefficients have been retained. We obtained a better result with the HDWT than with the SWT and HWT, in terms of PSNR and visually. As expected, textures are better preserved. Statistical comparisons are given in Figure 10. When few coefficient are retained the HDWT gives a much better approximation than the SWT. The two lines merge when more than 10 5 coefficients are retained. However the SWT never outperforms the HDWT contrary to what happens with redundant frames such as curvelets and contourlets.

As alluded to in the introduction, the benefit of directional transforms is larger in regions where the image has a strong and smooth geometrical flow. The Barbara image is therefore very well adapted to the HDWT. When the comparison is made on texture-less images, the improvement becomes less significant. For instance, when the 3 transforms are applied to the 512 × 512 fresco image and the 10 4 largest coefficients are retained, the PSNR of the obtained images are close (see Figure 11). We emphasize that the efficiency of the HDWT could be improved by designing new filter banks.

The proposed transform has however a drawback that appears clearly when we look at the detail shown in Figure 11. Although the HWT does not have ambiguous directions, it has only 3 directions. As a consequence, some directions are not well reproduced. With the configuration we adopted, this is the case for the horizontal structures that appears on the right of the image. As expected, only the horizontal and vertical structures are well preserved with the SWT. All the directions should be well represented with the HDWT. The left hand side of the image is better rendered with the HDWT than with the HWT and the SWT. But one can observe that the horizontal structures on the right hand side are not well preserved with the HDWT. This could be justified by the fact that, like the HWT, the HDWT has only 3 directions at the coarsest scales. However the 12 directions at the finest scales should make up for the lack of directions of coarse wavelets. The main reason for this phenomenon is that some directional wavelets are aliased. This problem is addressed in the next section. 4 On the frequency localization of Directional Wavelet Bases

Frequency localization of the HDWT

The aliasing alluded to above appears clearly on 6 of the wavelets shown in Figure 12 1 , in the case of 12 directions. More precisely, the 12 wavelets are aliased. Each of them has 2 directions, the main direction that corresponds to the ideal frequency support, and an aliased direction obtained by symmetry about the Cartesian axes. This phenomenon could be justified by the way the HDWT is implemented, that is to say by the combinations, at each scale, of an hexagonal filter bank and one or more 2-channel filter banks. This combination generates non-uniform filter banks [START_REF] Durand | M-band filtering and non-redundant directional wavelets[END_REF] that split the image into a refinement subband and, for example, 12 directional wavelet subbands as in Figure 8, and whose wavelet filters are aliased. Figure 13 shows the transfer functions of 2 among the 12 passband directional wavelet filters that are designed by the proposed method. Some spikes appear clearly in the right hand side graph. They do not appears in the left hand side one because the main direction and the aliased direction are too close. The causes for this frequency leaks are unfortunately not due to the way the 12 subbands are split. Suppose indeed that we want to compute directly the wavelet coefficients at a same scale, using a single non-uniform filter bank, and that we design therefore new filters. Then the obtained filters will be aliased again, for two reasons. First, the periodization of the transfer functions introduces energy in unwanted regions of the spectrum. It corresponds to the 2 largest peaks in the aliased direction, in Figure 13-(b). The same phenomenon appears in some other transforms based on filter banks such as decimated contourlets.

Another reason why the hexagonal directional wavelets are aliased is connected to the permissibility condition for the filter banks that are used in this transform. One says that a filter bank is (or more precisely the passband supports of its filters are) permissible if one can design smooth transfer functions whose supports remain in a neighborhood of their ideal supports [START_REF] Chen | Considerations in multidimensional filter bank design[END_REF], or in other words, if for all ǫ > 0, there exists a perfect reconstruction filter bank {H k } k such that supp H k ⊂ A k +B(0, ǫ), ∀k, where B(0, ǫ) stands for the ball of center 0 and radius ǫ, and A k is the ideal frequency support of the filter. Although each uniform (2-channel and 4-channel) filter bank used in the proposed transform is permissible, the non uniform filter banks generated by the combination of these filter banks are not permissible. The frequency support of the filter of Figure 13-(b) is schematized in Figure 14-(b). In Figure 14-(a) is shown the frequency support that would have a permissible filter if such a filter existed. Notice that by the periodization of the transfer function, the ideal filter is also aliased. A detailed proof of this property is given in [START_REF] Durand | M-band filtering and non-redundant directional wavelets[END_REF] for another frequency partitioning. It can be easily adapted to the HDWT. Moreover, the arguments developed in the proof of Proposition 2 below can be applied to show the non-permissibility of directional filter banks.

Frequency localization of general directional wavelets

From this argumentation, one might think that the obstruction to the construction of non-aliased directional wavelet bases could be bypassed if one uses wavelets that are not generated by a filter bank tree. (Recall that the periodization of transfer functions introduces aliasing.) Consider a Shannon directional wavelet basis {ψ s k ( •γ)} k∈K,γ∈Γ k that we want to approximate with regular wavelets. Let us suppose that the basis satisfies a parabolic scaling law so that the number of directions increases with the scale. Recall that this last property is fundamental to obtain an optimal sparsity behavior for images with C 2 singularities [START_REF] Candès | Curvelets -Multiresolution representation and scaling laws[END_REF]. The frequency support of each wavelet is adjacent to the support of 2 other wavelets that have the same scale and at least 1 wavelet that has a coarser scale. If the number of directions increases with the scale, there is also a wavelet whose frequency support is adjacent to the support of at least 2 wavelets that have a finer scale. There are therefore at least 5 neighboring wavelets. We claim that this Shannon wavelet basis cannot be approximated by a regular wavelet basis {ψ k ( • -γ)} k∈K,γ∈Γ k so that the frequency support of ψ k remains in an arbitrary small neighborhood of supp ψ s k , for all k ∈ K. We obtain this way a generalization of the property on the non-permissibility of the filter banks, discussed above.

Proposition 2 Consider a Shannon directional wavelet basis {ψ

s k ( • -γ)} k∈K,γ∈Γ k with ψ s k = |A k | -1/2 χ A k
, such that one of the sets A k has more than 4 adjacent sets A l . For ǫ > 0 sufficiently small, there are no families of functions {ψ k } k∈K and { ψ k } k∈K that satisfy the following properties simultaneously.

i) {ψ k ( • -γ)} k∈K,γ∈Γ k and { ψ k ( • -γ)} k∈K,γ∈Γ k are biorthogonal bases for L 2 (R 2 ). ii) ψ k and ψ k are continuous, ∀k ∈ K. iii) supp ψ k ⊂ A k +B(0, ǫ diam(A k )) and supp ψ k ⊂ A k +B(0, ǫ diam(A k )), ∀k ∈ K.
We tried to give a statement that applies to the most general frequency partitionings. In the proof, we consider 6 wavelets only, and in condition (iii), the radius of the ball does not have to depend on the diameter of A k . In the case when

A k = D -1 k A, it is however natural to assume that supp ψ k ⊂ D -1 K (A + B(0, ǫ)).
Proof. The proof by contradiction of this statement is based on geometric arguments. Suppose Conditions i),ii) and iii) are satisfied for an ǫ > 0 arbitrarily small. Denote by A 0 the frequency support of a Shannon wavelet having at least 5 neighboring wavelets, and by A k , k ∈ {1, • • • , 5}, the 5 neighboring wavelet frequency supports (see Fig. 15). The orthogonality condition

ψ 0 ( • -γ), ψ k = 0, ∀γ ∈ Γ 0 , ∀k ∈ {1, • • • , 5}
becomes, after a standard calculation,

γ∈Γ * 0 ψ 0 ( • -γ) ψ k ( • -γ) = 0, ∀k ∈ {1, • • • , 5}, (4)
while the condition leads to

ψ 0 ( • -γ), ψ 0 = 1 if γ = 0, 0 if γ ∈ Γ 0 \ {0},
γ∈Γ * 0 ψ 0 ( • -γ) ψ 0 ( • -γ) = 1, (5)
where z stands for the conjugate complex number of z. If Conditions ii) and iii) are satisfied for ǫ small enough, then the support of ψ 0 ψ k has 2 connected components. They are represented by the hatched areas in Figure 15, when k = 1. By (4), this 2 components should cancel each other out when ψ 0 ψ k is shifted along Γ * 0 . The reciprocal lattice Γ * 0 must also satisfy [START_REF] Candès | Curvelets -a suprizingly effective nonadaptive representation for objects with edges, Curve and Surface Fitting[END_REF] which means that the ideal support of ψ 0 forms a tiling of R 2 when it is shifted along Γ * 0 . These last 2 conditions are summed up in Figure 16-(a). We can observe that, in this example, the 2 components cancels each other out when k = 1 but does not when k = 2 (see Figure 16-(b)).

More generally, there does not exists a tiling {A 0 + {γ}} γ∈Γ * 0 for the ideal support of ψ 0 such that the 5 (or more) neighboring wavelets ψ k satisfy Condition (4) in the sense that the 2 connected components of ψ 0 ψ k can cancel each other out. Indeed, the tiling generated by the support of ψ 0 is an isohedral tiling [START_REF] Grünbaum | Tilings and paterns: an introduction[END_REF]. (It is generated by a single tile and its symmetry group acts transitively on the tiles.) Among the 81 types of possible isohedral tilings, several ones have tiles with more than 4 adjacent tiles but one can verify that none of them has the required type. An easy way to check this claim is to consider a tile formed by 2 adjacent tiles, as for instance, the tile marked by 0 in Figure 16. When it is shifted along the good regular lattice, it generates another isohedral tiling. If it has the required type, the tile should have at least 8 adjacent tiles as in Figure 17. The conclusion comes from the fact that a tile of an isohedral tiling has at most 6 adjacent tiles [START_REF] Grünbaum | Tilings and paterns: an introduction[END_REF].

⋄ 0 0 0 0 Figure 16: Ideal support of ψ 0 ( • -γ) for γ ∈ Γ * 0 .
The 2 components of the support of ψ 0 ( •γ) ψ 1 ( •γ) cancel each other out while the components of the support of Proposition 2 does not apply to the frequency partitioning of Figure 3-b as each set A k has only 4 adjacent sets. The argumentation above can however be generalized to this example if one includes sets that are connected by a dot, though the aliasing might remain localized in the neighborhood of a dot. This phenomenon occurs with hexagonal wavelets but that the artifact is visually negligible (see Figure 12-(a)). Notice that there are also non-aliased bases that satisfy a non isotropic scaling law but that are not obtained by shifting a finite number of mother wavelets at each scale. One might think here of orthonormal ridgelets [START_REF] Donoho | Orthonormal ridgelets and linear singularities[END_REF].

ψ 0 ( • -γ) ψ 2 ( • -γ) do not.

Directional Wavelet Tight Frames With Arbitrary Small Redundancy Factor

Proposition 2 reveals an obstruction to the construction of directional wavelet bases that approximate properly Shannon directional wavelets. On the other hand, second generation discrete curvelets [START_REF] Candès | Fast Dicrete Curvelet Transform[END_REF] approximate the associated Shannon wavelet tight frame with a good frequency localization. This frame is however highly redundant. We show in this section that the latter construction can be generalized in order to obtain well localized directional wavelet tight frames with arbitrary small redundancy factor.

Shannon Directional Wavelet Tight Frames

Proposition 1 can be generalized to the case of tight frames as follows.

Proposition 3 Let ψ s k = (1/|A k |) 1/2 χ A k with A k ⊂ R 2 , for all k ∈ K. If 1. {A k + {γ}} γ∈Γ * k is a packing of R 2 , 2. {A k } k∈K is a tiling of R 2 , 3. |C k | |A k | = ρ (≥ 1), ∀k ∈ K,
where |C k | is the area of the reciprocal cell of Γ k , then the family {ψ s k ( •γ)} k∈K,γ∈Γ k is a tight frame for L 2 (R 2 ) with redundancy factor ρ.

Proof. Condition 1 of Proposition 1 has been changed in Conditions 1 and 3. They are sufficient conditions for {ψ s k ( •γ)} k∈K,γ∈Γ k to be a tight frame for W k with redundancy factor ρ. ⋄

Regular Directional Wavelet Tight Frames

In the discussion below, we assume that the functions ψ k have fast decay and are bandlimited. We make therefore the assumption that ψ k is in the Schwartz class S(R 2 ) although Proposition 4 could be proved with more general functions. This hypothesis makes however convergence questions easier to treat.

Proposition 4 Suppose that ψ k ∈ S(R 2 ), ψ k 2 = 1, ∀k ∈ K, k∈K |C k | ψ k 2 ∞ < ∞ and min |x -y| : x, y ∈ k∈K Γ * k & x = y > 0.
The family {ψ k ( •γ)} k∈K,γ∈Γ k is a tight frame for L 2 (R 2 ) with redundancy factor ρ if and only if a.

k∈K

|C k | ψ k (ξ) 2 = (2π) 2 ρ, b. k∈Kγ |C k | ψ k (ξ -γ) ψ k (ξ) = 0, ∀γ ∈ k∈K Γ * k ,
where

K γ = {k ∈ K : γ ∈ Γ * k }.
Notice that the condition min{|x -y| : x, y ∈ k∈K Γ * k & x = y} > 0 is not satisfied by directional wavelets that are generated by standard rotations and dilations as considered in [START_REF] Antoine | Two-dimensional directional wavelets and the scale-angle representation[END_REF] and shown in Figure 3-(b). In this case, another proposition needs to be proved. On the other hand, this condition is satisfied by all the directional wavelets that fulfill a parabolic scaling law and such that low frequencies are represented by a scaling function φ. It is therefore satisfied by curvelets.

Proof. Let us assume that {ψ k ( •γ)} k∈K,γ∈Γ k is a tight frame for L 2 (R 2 ) with redundancy factor ρ. It is characterized by the equation

ρf (x) = k∈K γ∈Γ k f, ψ k ( • -γ) ψ k (x -γ), (6)
for all f ∈ L 2 (R 2 ) and almost every x ∈ R 2 , with unconditional convergence in L 2 (R 2 ). Applying a Fourier transform to this equation, we get

ρ f (ξ) = k∈K γ∈Γ k f, ψ k ( • -γ) e -iγ•ξ ψ k (ξ), for almost every ξ ∈ R 2 , with f, ψ k ( • -γ) = F f ψ k (-γ),
where F denotes the Fourier transform. Hence

ρ f (ξ) = (2π) -2 k∈K γ∈Γ k F f ψ k (-γ) e -iγ•ξ ψ k (ξ).
Using Poisson summation formula

γ∈Γ k ĝ(γ) e iγ•ξ = |C Γ k | γ∈Γ * k g(ξ -γ),
we obtain, let us say for all f ∈ S(R 2 ),

ρ f (ξ) = (2π) -2 k∈K |C Γ k | γ∈Γ * k f (ξ -γ) ψ k (ξ -γ) ψ k (ξ). Notice that k∈K |C Γ k | γ∈Γ * k f (ξ -γ) ψ k (ξ -γ) ψ k (ξ) ≤ γ∈Γ * k f (ξ -γ) k∈K |C Γ k | ψ k (ξ -γ) 2 1/2 k∈K |C Γ k | ψ k (ξ) 2 1/2 ≤ C < ∞
Then, by Fubini's Theorem,

ρ f (ξ) = (2π) -2 γ∈ k∈K Γ * k f (ξ -γ) k∈Kγ |C Γ k | ψ k (ξ -γ) ψ k (ξ). (7)
Since min{|x -y| : x, y ∈ k∈K Γ * k & x = y} > 0, ( 7) is satisfied, for all f ∈ S(R 2 ), if and only if Conditions a and b are satisfied.

Conversely, assume that Conditions a and b are satisfied. Equation ( 7) is therefore satisfied for all f ∈ S(R 2 ). The same arguments as above lead to [START_REF] Candès | Curvelets -Multiresolution representation and scaling laws[END_REF]. Now, suppose that f ∈ L 2 (R 2 ). By the density of S(R 2 ) in L 2 (R 2 ), there is a sequence {f n } n of functions in S(R 2 ) that tends to f . Every f n satisfies (6) and consequently

ρ f n 2 2 = k∈K γ∈Γ k | f n , ψ k ( • -γ) | 2 . (8)
By Fatou's Lemma, we know therefore that k∈K γ∈Γ

k | f, ψ k ( • -γ) | 2 converges.
It is then easy to prove that

ρ f 2 2 = k∈K γ∈Γ k | f, ψ k ( • -γ) | 2 ,
for all f ∈ L 2 (R 2 ), by making f n tends to f in [START_REF] Cohen | Biorthogonal bases of compactly supported wavelets[END_REF]. Hence the conclusion. ⋄

Below, we will need only the following sufficient conditions for a tight frame.

Corollary 1 Suppose ψ k 2 = 1, ∀k ∈ K and a.

k∈Kγ

|C Γ k | | ψ k (ξ)| 2 = (2π) 2 ρ, b. supp ψ k ( • -γ) ∩ supp ψ k = ∅, ∀k ∈ K, ∀γ ∈ Γ k , then {ψ k ( • -γ)} k∈K,γ∈Γ k is a tight frame for L 2 (R 2
) with redundancy factor ρ.

A consequence of this result is that, for all ρ > 1, there is a directional wavelet tight frame with redundancy factor ρ, and such that, for all k ∈ K, ψ k has fast decay and ψ k is supported on a neighborhood of its ideal support. Her is an example that prove this claim. The idea is to construct a Shannon directional wavelet tight frame with arbitrary small redundancy factor ρ > 1 using Proposition 3. If d(A k + {γ}, A k + {η}) > 0, for all γ, η ∈ Γ * k with γ = η, and all k ∈ K, where d stands for the distance, then the wavelets ψ k are obtained by smoothing χ A k in such a way that, supp ψ k ( •γ) and supp ψ k ( •η) are disjoint for all γ, η and k. Condition a of Corollary 1 can also easily be satisfied in order to construct the sought regular tight frame.

{ψ k ( • -γ)} k∈K,γ∈Γ k 0 0 (a) (b)
The set A = {ξ ∈ R 2 : α|ξ 2 | < |ξ 1 | and 1 < |ξ 2 | < 2} considered in Section 2 cannot be used any longer. Indeed, if it is shifted along the same lattice as in Figure 3-(a), the smoothed Fourier transforms of the wavelets cannot have disjoint supports. There are, on the other hand, lattices Γ * such that {A + {γ}} γ∈Γ * is a packing of R 2 and d(A k + γ, A k + η) > 0 if γ = η, but their density cannot be arbitrary close to 1. An example of such a packing is given in Figure 18-(a). (This kind of packing is used in the construction of curvelets [START_REF] Candès | Fast Dicrete Curvelet Transform[END_REF].)

Consider henceforth the set

A = {ξ ∈ R 2 : |ξ 1 | < α|ξ 2 | and 1 < |ξ 2 | < β} with β ∈ (1, 2
). When it is shifted along Γ * = Ze 1 + Ze 2 where e 1 = (0, (β + 2)/2) and e 2 = (α(β + 1)β + 2, (β + 2)/4), it forms a packing which density tends to 1 as β tends to 2. This example is shown in Figure 18-(b). We show in Figure 19-b, the frequency partitioning corresponding to the obtained β-adic directional wavelets. Notice that in the limit case when β = 2, we find again the tiling of Figure 3-(a). It is, of course, not realistic to choose β too close to 2 as, in such a case, the wavelets have no aliasing and fast decay but, by Heisenberg principle, their essential spatial support is too large (and the wavelets are therefore not sufficiently localized in the spatial domain). Moreover the discretization process needed for an implementation on images with a finite number of pixels, does not permit to make β tends to 2 either. Nevertheless, we can expect to obtain properly localized (in space and frequency) wavelets tight frames with redundancy factor less than 2 while curvelets have a redundancy factor equal to 7.2 [START_REF] Candès | Fast Dicrete Curvelet Transform[END_REF].

Conclusion and outlook

The new directional wavelet transform that we proposed in Section 3 is implemented in a simple and fast algorithm. In some sense, it improves the curvelet and contourlet transforms as it is non redundant. However it has lost in frequency localization since the directional wavelets are aliased. In spite of this drawback, we have obtained encouraging numerical results. The later could be improved by developing new filter design techniques. In particular, FIR filters could be used when the transform is applied to hexagonally sampled images, and would permit to reduce the Gibbs and aliasing phenomenon.

Proposition 2 shows that we cannot construct directional wavelets (as they are defined in this article) without aliasing or redundancy. The proposed transform can therefore be very attractive when we cannot manage with redundant transforms as, for instance, in lossless compression. On the other hand, we shown in Section 5 that the redundancy factor of well localized directional wavelet tight frames can be dramatically reduced, and consequently that these wavelets could be used in low bit rate compression.

The discussion above leaves however some open questions. In particular, we shown that there are no biorthogonal directional wavelet bases that are both well localized in the sense that the wavelets and the dual wavelets have fast decay and their Fourier transform are supported on an arbitrary close neighborhood of their ideal support. But we do not know whether there are well localized directional wavelet bases whose dual basis is aliased. These transforms would be very useful since images could be reconstructed without aliasing artifact by reserving the well localized wavelet bases for the synthesis.

For instance, consider a Shannon directional wavelet basis given by the sets {A k } k∈K and the lattices {Γ k } k∈K , with A k = D * k A and Γ k = D -1 k Γ except possibly for one index k corresponding to the low frequency wavelet φ. Define ψ by its Fourier transform

ψ(ξ) = 2π|A| -1/2 g * χ A (ξ) e iα•ξ , ∀ξ ∈ R 2 ,
where g is supported on a neighborhood of 0, R 2 g(x) dx = 1 and α ∈ Γ so that the wavelets are not too much aligned. It satisfies the identity

γ∈Γ * ψ( • -γ) 2 = (2π) 2 |A| ,
which means that {ψ( •γ)} γ∈Γ is an orthonormal basis for W = ψ( •γ) γ∈Γ . More generally, if ψ k = ψ • D k , then {ψ k ( •γ)} γ∈Γ k is an orthonormal basis for W k = ψ k ( •γ) γ∈Γ k , but we do not know whether {ψ k ( •γ)} k∈K,γ∈Γ k is a basis for L 2 (R 2 ). In particular, do we have

L 2 (R 2 ) = k∈K W k ?
7 Appendix: Fast Fourier Transform on regular lattice.

For convenience, we give a brief description of a fast algorithm for the computation of the Fourier transform of an image I defined on a regular quotient lattice Γ/Λ with Γ ⊂ Λ ⊂ R 2 . The Fourier transform of I is defined by

I[λ] = γ∈Γ/Λ I[γ] e -i γ,λ , ∀λ ∈ Γ * /Λ * .
Notice that the inverse Fourier transform is given by the reconstruction formula

I[γ] = 1 #(Γ/Λ)

Figure 1 :

 1 Figure 1: (a) Detail of an image. (b) Its spectrum is concentrated along a line.

Figure 2 :

 2 Figure 2: Frequency support of (a) Wavelets, (b) Bandelets (first generation), (c) Curvelets.

-

Figure 3 :

 3 Figure 3: Examples of tilings (a) {A + {γ}} γ∈Γ and (b) {A k } k∈K .

Figure 4 :

 4 Figure 4: (a) Hexagonal filter bank and (b) ideal frequency supports of hexagonal wavelets (↓ 4 stands for keeping the points of 2Γ only).

Figure 6 :

 6 Figure 6: Ideal frequency supports of hexagonal directional wavelets.

Figure 7 :

 7 Figure 7: Two possible reciprocal cells for the square lattice, (a): Voronoi cell and (b): irregular hexagonal cell. Periodization of a square reciprocal cell, (a): square lattice. (c) : irregular hexagonal lattice. The dashed lines corresponds to the Voronoi set of the hexagonal reciprocal lattice. The black disk is the essential frequency support of an image.

Figure 8 :

 8 Figure 8: Decomposition of the Barbara image onto 12 directional subbands.

Figure 9 :

 9 Figure 9: Non linear approximation of Barbara image (detail). (a) Original image. (b) HWT (PSNR = 27.08). (c) SWT (PSNR = 27.02). (d) HDWT (PSNR = 28.58). Images are reconstructed from the 10 4 largest coefficients.

Figure 10 :

 10 Figure 10: PSNR versus number of retained coefficients for Barbara image (dashed line: SWT, solid line: HDWT).

Figure 11 :Figure 12 :

 1112 Figure 11: Non linear approximation of fresco image (detail). (a) Original image. (b) HWT (PSNR = 24.85). (c) SWT (PSNR = 24.87). (d) HDWT (PSNR = 25.03). Images are reconstructed from the 10 4 largest coefficients. The 3 images are aliased but in different directions

Figure 13 :Figure 14 :

 1314 Figure 13: Transfer functions of 2 directional wavelet filters.

Figure 15 :

 15 Figure 15: Ideal frequency support of ψ k for k ∈ {0, • • • , 5}. The hatched areas correspond to the support of ψ 0 ψ 1 .

Figure 17 :

 17 Figure 17: The required type of tiling does not exist since tiles of an isohedral tiling have at most 6 adjacent tiles.

Figure 18 :

 18 Figure 18: Packings {A + {γ}} γ∈Γ for (a) curvelets and (b) directional wavelets with dilation factor < 2.

Figure 19 :

 19 Figure 19: Packings {A k } k∈K for (a) curvelets and (b) directional wavelets with dilation factor < 2.

3 i=0( 3 i=0I 3 i=0(

 333 λ∈Λ * /Γ * I[λ] e i γ,λ , ∀γ ∈ Λ/Γ, so that the FFT and the IFFT can be computed using a similar scheme. Let us show that they are implemented in a O(N log N ) algorithm where N = #(Γ/Λ). Suppose first that 2Γ ⊂ Λ and write Γ asΓ = 2Γ + {γ i }) ,with γ i ∈ Γ, ∀i ∈ {0, • • • , 3} (and γ 0 = 0). Thus we haveI[λ] = 3 i=0 γ∈Γ/(2 -1 Λ) I[2γ + γ i ] e -i γ,2λ e -i γi,λ = i [2λ] e -i γi,λwhereI i = (γ → I[2γ + γ i ]) is defined on Γ/(2 -1 Λ) with #(Γ/(2 -1 Λ)) = #(Γ/Λ)/4.So, to compute I, one just need to compute the Fourier transform of the 4 smaller images I i and multiply them by e -i γi,λ .Suppose that #(Γ/Λ) = 4 n and denote by C n the computation cost of I. If one neglects the cost of sums, we obtain the recurrence relationC n = 4 C n-1 + 4 n .(Since γ 0 = 0, we have actually C n = 4 C n-1 + 3 4 n-1 .) We can decrease n until 2Γ ⊂ Λ. In such a case, we find another lattice ∆ ⊂ Λ such that Γ = ∆ + {γ i }) , and apply the same process. As C 0 = 0, we get C n = 4 n n. As a conclusion, for an image of size N = 4 n , the computation cost is N log 4 N .

We do not show the discrete wavelets at the finest scale of the HDWT, but the wavelets defined on a continuous domain


Pierre Savagner
email: pierre.savagner@inserm.fr

Thomas Brabletz

Chonghui Cheng

Christine Gilles

Tian Hong

Myriam Polette

Guojun Sheng

Marc P Stemmler

Erik W Thompson

Twenty years of EMT: A state of the field from TEMTIA X

Keywords: epithelial-mesenchymal transition, development, cancer

Jean Paul Thiery, and saw the implementation of the Betty Hay Award (currently AU$1,000) for newly independent female EMT researchers, and the Betty Hay Oration, a Plenary lecture, both awarded at each subsequent meeting. An evolving constitution of TEMTIA has progressively incorporated an executive Committee and an advisory committee. A dedicated website (www.TEMTIA.org) was set up, providing announcements and information and preparing the organization for the challenges of the next decades. Recently a consensus statement was published on behalf of TEMTIA to provide guidelines for EMT-related definitions [START_REF] Yang | Guidelines and definitions for research on epithelial-mesenchymal transition[END_REF].

Following this retrospective, scientific talks were distributed in sessions focusing on specific current and emergent aspects of both cancer-associated and physiological EMTs.

Metabolism meets EMT

One emerging topic in the domain of cell plasticity is the link between metabolism and cell phenotype. Several examples were provided in a dedicated session. First, Sarah Fendt (KU Leuven, Belgium) discovered an intriguing integrin modulation in cancer cells deprived of phosphoglycerate dehydrogenase, resulting in early cancer cell dissemination. Phosphoglycerate dehydrogenase expression pattern was found to be a marker for metastasis occurrences. More generally, she is currently investigating nutrient dependencies in metastases and exposing organ specific features [START_REF] Bergers | The metabolism of cancer cells during metastasis[END_REF].

Another speaker, Raphael Rodriguez (Institute Curie, Paris, France) focused on a new role for CD44 glycoprotein in mediating copper and iron uptake by the cell. These metals were found to regulate cell plasticity through mechanisms that could be targeted by new classes of drugs, found to inhibit EMT in cancer cells [START_REF] Müller | CD44 regulates epigenetic plasticity by mediating iron endocytosis[END_REF]. Bob Weinberg (MIT, Cambridge, USA) elaborated in this direction, singling out ferroptosis as a new target, preferentially impacting mesenchymal-like carcinoma cells. Similarly, a very distinct pathway, involving short fatty-acids including propionate was found by P. Ceppi (U Southern Denmark, Denmark) to reduce EMT episodes in non-small cell lung cancer, as well as metastases. The pathway involved chromatin configuration and histone acetylation, a more usual component of EMT pathways (accepted for publication in EMBO molecular medicine journal).

Overall, these talks emphasized the very comprehensive meaning of the EMT process, defining a whole new cell identity, in structural and functional aspects during developmental stages [START_REF] Bhattacharya | Neural crest metabolism: at the crossroads of development and disease[END_REF] as well as cancer progression.

New developmental and physiological perspectives

The EMT process was first described in vivo during specific stages of development that strongly contributed to establish EMT archetypical characteristics. One revealing observation is the increasing number of developmental stages in which an EMT represents a necessary step. Several new cases were detailed during the developmental sessions, starting with the EMT implicating mesothelial cells giving rise to the chorioallantoic membrane, a critical stage during placental development (Guojun Sheng, Kumamoto University, Japan). Also in mammals, Vincent Guen (CRCI2NA, Nantes, France)

showed how an EMT stage affecting mammary stem cells involves primary ciliogenesis as a necessary catalyzer for stemness and potentially tumorigenesis [START_REF] Guen | EMT programs promote basal mammary stem cell and tumor-initiating cell stemness by inducing primary ciliogenesis and Hedgehog signaling[END_REF].

The role of MMP during Xenopus neural crest cell emergence from a classic EMT was revisited by Eric Théveneau (Center for Integrative Biology, CNRS, France), showing an early and non-canonical role for MMP28 [START_REF] Gouignard | Paracrine regulation of neural crest EMT by placodal MMP28[END_REF]. In drosophila, Kyra Campbell (University of Sheffield, UK) exposed the role of Serpent/GATA 4/6 in coordinating cell reconfiguration and deconstructing EMT functional constituents using multiple molecular and optical approaches [START_REF] Campbell | Differential roles of the Drosophila EMTinducing transcription factors Snail and Serpent in driving primary tumour growth[END_REF].

Endothelial cells (EC) represent a specific class of epithelial cells lining up blood vessels. They can undergo an endothelial-mesenchymal transition (EndoMT) during developmental (angiogenesis, regeneration) and pathological (wound healing, atherosclerosis) events. Chris Hughes (UC Irvine, CA, USA) analyzed the role of Slug/Snai2 in angiogenic sprouting in vitro, characterizing Notch family members and specific effectors modulating EC-pericyte interactions and describing increasing

EndoMT degrees [START_REF] Hultgren | Slug regulates the Dll4-Notch-VEGFR2 axis to control endothelial cell activation and angiogenesis[END_REF]. Jatin Patel (QUT, Brisbane, Australia) also studied EndoMT pathways in mouse and human endothelium, in link with fibrosis and atherosclerosis. He described new vessel-resident stem cells activating alternatively Rbpj and Sox9 to modulate an EndoMT process [START_REF] Zhao | Sox9 and Rbpj differentially regulate endothelial to mesenchymal transition and wound scarring in murine endovascular progenitors[END_REF]. These studies collectively provided new avenues for therapeutic targeting in the vasculature.

EMT staging and dynamics in cancer

The EMT process is well known to participate to cancer progression and metastasis. However the lack of stem cell potential in cell expressing a fully mesenchymal phenotype, found to be shortlived [START_REF] Fumagalli | Plasticity of Lgr5-Negative Cancer Cells Drives Metastasis in Colorectal Cancer[END_REF]. In a distinct genetic mouse model of skin squamous cell carcinoma, Evgenia Pastushenko iv sit i d B u s, Belgium) identified 6 tumor cell groups associated with various EMT stages. Taking advantage of the expression of Netrin1 by the most mesenchymal tumor cell group, she found that anti-Netrin1 antibodies sensitized tumor cells to chemotherapy, a promising preclinical direction for therapeutic targeting [START_REF] Pastushenko | EMT Transition States during Tumor Progression and Metastasis[END_REF].

EMT pathways in tumor stroma and links to immune response

The stiffness and nature of the tumor stroma directly impact tumor cells, as exposed by Danijela Vignjevic Matic (Institute Curie, Paris, France). Combining genetic and physical manipulations in vivo with microfabrication and force measurements, in vitro she showed the stiffness impact on tumor cells, inducing Yap delocalization and actin/myosin contractility, functional precursors for an EMT [START_REF] Clark | Self-generated gradients steer collective migration on viscoelastic collagen networks[END_REF].

More generally, EMT processes are associated with several families of transcription factors, notably the Snail, Twist and Zeb families, found to be expressed in all sensu stricto EMT cases during development, presumably in response to stroma signals. Jing Yang (U California, San Diego, La Jolla CA, USA) thus modulated stiffness in 3D mammary organoids by artificial means. She found Twist1 to be a key player, in partnership with the G3BP2/EphA2/LYN complex in transmitting mechanical cues to the tumor cells [START_REF] Fattet | Matrix rigidity controls epithelial-mesenchymal plasticity and tumor metastasis via a mechanoresponsive EPHA2/LYN complex[END_REF]. These factors are also linked to cell stemness. In the mammary carcinoma context, Chrysoula Tsirigoti (Uppsala University, Sweden) found that Snail-deficient tumor cells gained stemness potential and could develop differentiated acinar mammospheres in 3D culture, overexpressing the transcription factor FOXA1 and evoking the intermediary luminal progenitor-like phenotype described in triple-negative breast tumors [START_REF] Tsirigoti | Loss of SNAI1 induces cellular plasticity in invasive triple-negative breast cancer cells[END_REF].

Novel aspects of EMT-TF function came from observations that their expression is not limited to tumor cells and were also found to be upregulated In tumor stroma cells. Marc Stemmler (Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany) exploited a mouse model for colorectal cancer to find that Zeb1 expression drives cell plasticity also in cancer-associated fibroblasts in favor of tumor progression and immune evasion [START_REF] Schuhwerk | ZEB1-dependent modulation of fibroblast polarization governs inflammation and immune checkpoint blockade sensitivity in colorectal cancer[END_REF].

This link between EMT and immune evasion is now widely acknowledged and was explored by other speakers. Ben Stanger (U Pennsylvania, Philadelphia PA, USA) described a tumor cell resistance to T cell killing independent of antigen presentation but relying on immune checkpoints, mediated by IRF6 in a mouse pancreatic ductal adenocarcinoma model [START_REF] Kim | Plasticity-induced repression of Irf6 underlies acquired resistance to cancer immunotherapy[END_REF]. Anushka Dongre (Cornell University, Ithaca NY, USA) has analyzed these links in mouse mammary tumors models, showing distinct lymphocyte and macrophage recruitment according to tumor cell phenotype. The expression of CD73 in more mesenchymal tumors was linked to a decreased CD8 cytotoxic and CD4 helper T-cell infiltration, accompanied by a poor response to immune checkpoint blockade [START_REF] Dongre | Direct and Indirect Regulators of Epithelial-Mesenchymal Transition-Mediated Immunosuppression in Breast Carcinomas[END_REF]. Stéphane Terry (Gustave Roussy, INSERM U1186, France) described how hypoxia promotes EMT, CTL mediated killing and immune resistance in non-small cell lung cancer model through the TGF pathway, but also through AXL-mediated resistance to NK and CTL-mediated killing, suggesting therapeutic opportunities [START_REF] Terry | New insights into the role of EMT in tumor immune escape[END_REF]. Finally, Jonathan Kurie (U Pennsylvania, Philadelphia USA) described the connection between EMT and a Golgi-dependent hypersecretory process that drives pro-metastatic tyrosine kinase receptor burying inside intracellular vesicles, in link with immunosuppression in the tumor microenvironment [START_REF] Tan | EMT-activated secretory and endocytic vesicular trafficking programs underlie a vulnerability to PI4K2A antagonism in lung cancer[END_REF].

Comprehensive -omics approaches to EMT

Recent years have seen a surge in high-throughput multi-omics strategies to study tumor heterogeneity and dynamics, a key consequence of EMT process. They brought unprecedented understanding on tumor microenvironment and helped bringing the EMT concept closer to the clinical field. With this goal, Karuna Ganesh (Memorial Sloan Kettering Cancer center, New-York USA) has combined transcriptomics, epigenomics and spatial/histological approaches with composite tumor organoids to analyze colorectal cancer metastasis, unveiling relevant signaling pathways involved in metastasis [START_REF] Moorman | Progressive Plasticity during colorectal cancer metastasis[END_REF]. To link the power of single cell analysis to the dynamics of EMT phases, Jianhua Xing (U Pittsburgh, PA USA), designed the dynamo framework to build vector fields predicting cell fate and underlying regulation by combining single cell RNA seq with RNA velocity.

Such analysis was applied in EMT cell models to reveal distinct cell transition paths involving stops at G1/S or G2/M phase during cell cycle [START_REF] Qiu | Mapping transcriptomic vector fields of single cells[END_REF]. The role of chromatin conformation changes in such transitions was examined at a large scale by Ruby Huang (National Taiwan University, Taiwan), using

Hi-C analysis and ChIP-seq data from cancer cell lines. Her approach emphasizes the importance of topological remodeling for EMT-linked genes, with a specific enrichment in H3K27me3 in epithelial gene-residing topologically associated domain [START_REF] Pang | 3D genome organization in the epithelial-mesenchymal transition spectrum[END_REF]. Also based on a cell line-based EMT model, Joseph Taube (Baylor University, TX USA) studied chromatin accessibility dynamics to show the progressive nature of the E-cadherin gene repression. EMT process was marked with an increase in global chromatin accessibility, associated with the suppression of the chromatin looping factor CTCF.

This configuration supported the stability of an induced EMT hybrid phenotype [23]. An EMT-MET PHENOtypic STAte MaP (Phenostamp landscape) approach using a mass cytometry time course analysis after TGF treatment (EMT) then withdrawal (partial MET), cytoff single cell massspectroscopy panel of EMT markers was described by Loukia Karacosta (MD Anderson Cancer Centre, Houston USA), with an emphasis on MUC-1, a novel target in the MET reversal [START_REF] Karacosta | Mapping lung cancer epithelial-mesenchymal transition states and trajectories with single-cell resolution[END_REF].

New EMT targets for cancer therapy

Oncologists have followed the EMT field progress years after years, waiting for new targets and new drugs. Recently new candidates targeting metastasis as well as immune response (see above) are being tested in clinical trials by several hospitals and drug companies. Jean Paul Thiery (Guangzhou laboratory, Guangzhou, China), a founder of the EMT concept and pioneer of targeting efforts, described his strategy to specifically target EMT process, as a complementary scheme to targeted treatments [START_REF] Voon | The EMT spectrum and therapeutic opportunities[END_REF]. With the same goal, the drug company Transcenta developed an antibody targeting Gremlin 1, a BMP antagonist member of the TGF superfamily and an EMT inducer expressed by cancer associated fibroblasts as well as tumor cells. This antibody has demonstrated anti-tumor activity in prostate cancer and is now being tested in a global Phase 1 study [START_REF] Cheng | Gremlin1 is a therapeutically targetable FGFR1 ligand that regulates lineage plasticity and castration resistance in prostate cancer[END_REF]. More generally, drug companies have shown an increase of interest for drugs targeting EMT phases, especially in the context of personalized medicine. Some specific targets were mentioned throughout the talks, such as a gastrin-related peptide receptor identified in cells undergoing EMT (Veronique Delmas, Institute Curie, Paris, France) [START_REF] Raymond | Targeting GPCRs and Their Signaling as a Therapeutic Option in Melanoma[END_REF], Netrin-1 and FAT-1 (Ievgenia Pastushenko, Brussels University, Belgium [START_REF] Lengrand | Pharmacological targeting of netrin-1 inhibits EMT in cancer[END_REF]. Another promising approach was presented by Dana Ishay-Ronan (Sheba Medical Center, Ramat Gan, Israel) exploiting tumor cell plasticity of the EMT program to transdifferentiate breast cancer cells into postmitotic adipocytes. Based on her findings, BRCA1 mutations influence the response to TGF which might help to stratify patients for specific differentiation therapies [START_REF] Bar-Hai | Engaging plasticity: Differentiation therapy in solid tumors[END_REF]. She also was awarded with the Betty Hay award for early career female scientists.

A

An evolutionary perspective for physiological EMT

As EMT and its context-specific peculiarities in higher order multicellular organisms like vertebrates are very complex, important general mechanistic insight into the process and its biological meaning can be derived from much more simple multicellular organisms. Of note, EMT stages are described in all pluricellular organisms, starting with the simplest forms. Classic EMT pathways including "classic" EMT-TF are already found in Cnideria, if not in sponges. Three speakers explored archaic animal models and pluricellular aggregates to question the biological meaning of EMT. Patrick Humbert (Latrobe Institute Medical Science, Melbourne, Australia) explored the establishment and modulation of cell polarity, early target of the EMT process, in Trichoplax, one of the simplest multicellular organisms, to identify original mechanisms, involving the Scribble polarity module. He then described a fascinating ongoing experiment involving Trichoplax cellular organization and polarization in the absence of gravity, in space [START_REF] Wright | Cell polarity signalling at the birth of multicellularity: What can we learn from the first animals[END_REF].

In the search of an evolutionary origin for EMT, we went to the edge of pluricellularity in primitive organisms such as Choanoflagellates. Thibaut Brunet (Institut Pasteur, Paris, France) compared colonial forms of this organism, involving cohesive polarized cells with an apical flagellum with temporary amoeboid forms resulting from a stress-induced switch (confinement...), activating myosin-based motility and individual migration, a common feature with post-EMT cells [START_REF] Brunet | Cell polarity in the protist-to-animal transition[END_REF]. This view of EMT as an ancestral escape mechanism for cells to leave an organized "epithelium" brings new perspective on cancer progression, but also on developmental occurrences of EMT, extremely conserved during evolution. This relationship between multicellularity and emergence of cellular properties [START_REF] Pradeu | Reuniting philosophy and science to advance cancer research[END_REF] was finally scrutinized in cancer using several models to study the breakdown of multicellularity in cancer cells (Bertrand Daignan-Fornier, IBGC, Bordeaux France). Such perspective might be beneficial to draw more generalized conclusions and extraction of mechanisms that can be exploited therapeutically.

In conclusion, major scientific progress has been seen in several areas over the past two decades, involving the strong relationship between evolving cell states and EMT, providing molecular definition of developmental and disease EMT. This is being achieved by combining high throughput multi-omic studi s, i c udi "si c " pp o ch s, d sp ti pp ci tio of c p sticity i both developmental and malignancy environments. This has also provided new understanding about the disputed role of the mesenchymal phenotype in metastasis and of EM hybrid cells, potentially better fit for the metastatic decathlon. These conceptual advances were also supported by more accurate mathematical modelling around the process of hysteresis and by the collaborative use of developmental systems for cancer studies. Much of this progression is chronicled in special issues of the Journal Cells Tissue Organs, formerly Acta Anatomica, from Karger Publishers, which remains a long-term partner of TEMTIA with support at each meeting.

Figure legend

TEMTIA, the

  story TEMTIA X opened on a historical retrospective of the 20 years of TEMTIA jointly prepared by initial co-fou d s Do N w d Rik Thompso which c vi w d i th "m m s" s ctio of th TEMTIA website. Founded to promote exchange between major disciplines affected by EMT (Development, Cancer, Pathology), TEMTIA was initiated (incorporated in NSW, Australia, 2001) with a heavily international 1 st meeting (Port Douglas, Australia, 2003) program guided by a prominent international committee led by EMT pioneer Prof Elizabeth (Betty) Hay. Subsequent meetings were held in America, Europe and Asia, with the next meeting planned for Seattle, USA, November 2024, led by Caroline Hookway, A I stitut fo B i Sci c . B tty H y's h th w s dy i d c i y th 2003 V couv meeting, and the 2005 saw heartfelt memorial presentations by close colleagues Raghu Kalluri and

  , it has become clear in the last few years that intermediate cell phenotypes are dominating the scene in most carcinoma. The rise of in vivo imaging techniques, single cell sequencing and other omics has brought powerful new tools linking phenotype and tumor cell response. Jacco Van Rheenen (NCI, Amsterdam, Netherlands) used intravital microscopy in a murine mammary tumor model to compare tumor cell lifespan, stemness potential and phenotype, demonstrating with live markers

Figure 1 :

 1 Figure 1: 2022 TEMTIA's quest for EMT took place in Paris, home for the 15th century unicorn tapestries (Cluny Museum, Paris), an evocation of EMT multiple faces observed through the mirror of TEMTIA, and a leitmotiv for the symposium.

  metastasis, survival, therapy resistance, and predictive markers -where EMT is important for directing chemotherapy / immunotherapy. More targeted work is needed to identify the best markers, in which tumor types, and using which techniques (IHC, CTC, ctDNA). Treatment wise, clinical panelists discussed the need for novel trials approaches such as window-of-opportunity trials and neoadjuvant trials, where new agents could be trialed in combination with standard of care therapies. The significant financial and regulatory hurdles were discussed and the need for partnership with Pharma. A noteworthy initiative is a dedicated effort since the 2019 Kumamoto meeting to focus attention and activity on clinical implementation of the abundant implications of EMT in cancer progression and therapy resistance. A special virtual meeting (TEMTIA 9.5; Targeting EMT) was held in December 2021, and remains a key collaborative focus of TEMTIA meetings.

	target (standard therapy-resistant) pM state, or others? This led mostly to a preference to selectively
	block plasticity, rather than pushing cells into states that may enhance other aspects of metastasis-
	associated EMP (e.g. plastistatic approaches, Berx), or selectively kill EM hybrid cells by targeting the
	unique co-expression of E and M molecules in the same cell (Thompson). Additional avenues
	included EM-Hybrid markers such as Netrin-1, FAT1 or MUC1, as mentioned in the talks.
	Perspectives were also discussed for diagnostic markers -where EMT defines subtypes (e.g. claudin-
	low type breast cancers / basal breast cancers), prognostic markers -where EMT defines clinical
	outcome such as

very active multidisciplinary panel led by Thomas Brabletz, including Rik Thompson (Queensland University of Technology -QUT, Australia), Geert Berx (Ghent University, Belgium), James Lorens (University of Bergen, Norway), Jonathan Kurie (MD Anderson Cancer Center, Texas, USA), Karuna Ganesh (Memorial Sloan Kettering Cancer Center-MSKCC, NY, USA), Pierre Savagner (INSERM/Gustave Roussy, France) and others discussed the perspectives for incorporating EMT and related epithelial mesenchymal plasticity (EMP) into therapy for clinical benefit. The lack of clinical translation was attributed to issues such as the biphasic nature of EMP in metastatic progression, with pro-EMT phases and pro-MET phases. Thomas Brabletz posed the question: Should the strategy be to prevent EMT, restore an Epithelial state, drive to Mesenchymal state, block plasticity, directly

Acknowledgement

We are thankful to the 90 scientists that contributed to the oral communications and poster sessions during the 3 days of the meeting TEMTIA X. We clearly did not have the space to mention here all the contributions and apologize to the non-cited authors for a subjective choice...

Funding Sources

This work was supported by the Ligue Nationale contre le cancer (grant: 2021R21078L ).

Statements

Conflict of Interest Statement

The authors have no conflicts of interest to declare.

Author Contributions

All authors contributed to the writing and/or editing of the manuscript. The organization and coordination were supervised by P. Savagner.