Tobias Isenberg
email: isenberg@cs.rug.nl

Position Paper: Touch Interaction in Scientific Visualization

Keywords:

Direct-touch interaction is receiving an increasing amount of attention in the HCI domain and has recently been applied to some problems in information visualization. However, the field of scientific visualization has not seen much work on how direct-touch interaction could benefit the interactive visualization process. One reason may be that scientific visualization typically deals with datasets that are defined in 3D space, while touch input is two-dimensional. Therefore, to control scientific datasets in interactive visualizations one has to define intuitive mappings from 2D input to 3D manipulations for a variety of data types and exploration techniques. In this position paper we discuss some of the challenges of using direct-touch interaction in the field of scientific visualization that arise from the specific constraints of scientific visualization and argue for the development of integrated techniques to overcome these problems.

INTRODUCTION

Scientific visualization is the science of creating graphical representations of scientific data that is the basis of research in virtually all domains of science. Scientific visualization (as opposed to information visualization) deals predominantly with data that is spatially explicit in 3D, i. e., for each data point a precise 3D location is known. Such data includes CT/MRI scans in medicine; particle simulations in physics, astronomy, or swarm behavior; molecular models in genetics, biology, chemistry, or material sciences, or shapes of functions or sets in mathematics, to name but a few examples.

It is a recognized fact that a good visualization needs to support user activities beyond viewing the data [START_REF] Ware | Information Visualization -Perception for Design[END_REF]. Scientists need to be able to drill down and find details about what seems important, relate information on many levels of granularity, and gain an encompassing picture about relationships and correlations present in their data to form hypotheses and plan next step actions. To support interactivity in scientific visualization two main aspects require dedicated research attention: (a) interaction models for control and exploration of the visualization to support problem solving and (b) interactive rendering speeds to achieve real-time refresh rates for minimal disruption of higher-level tasks. Many new techniques have been and are being developed based on GPU processing which address the second challenge successfully. The challenge of providing novel solutions for interactive exploration, navigation, and problem solving with threedimensional scientific visualizations [START_REF] Keefe | Integrating Visualization and Interaction Research to Improve Scientific Workflows[END_REF] is one that has received less attention in the field of scientific visualization to date. We therefore argue in this paper that we need to embrace emerging interactive display technology and inves-tigate its use for creating engaging and intuitive interactive next-generation scientific visualization work environments.

DIRECT-TOUCH INTERACTION

In particular, touch-sensitive displays have numerous important but unexplored benefits for scientific visualization: they provide enough area and resolution to explore visualizations, they facilitate awareness in collaborative settings [START_REF] Hornecker | Collaboration and Interference: Awareness with Mice or Touch Input[END_REF], and they offer natural direct-touch interaction which provides somesthetic information and feedback beneficial for effective interaction both in real and virtual environments [START_REF] Robles-De-La-Torre | The Importance of the Sense of Touch in Virtual and Real Environments[END_REF], and the direct-manipulation metaphor of touch interaction allows people feel in control of their data [START_REF] Isenberg | Studying Direct-Touch Interaction for 2D Flow Visualization[END_REF] and has shown to outperform mouse input for specific tasks [START_REF] Kin | Determining the Benefits of Direct-Touch, Bimanual, and Multifinger Input on a Multitouch Workstation[END_REF]. Despite these advantages, however, to date there is little support for interactive scientific visualization on these large displays. While touch interaction has previously been explored in a general visualization context (e. g., [START_REF] Forlines | DTLens: Multi-User Tabletop Spatial Data Exploration[END_REF][START_REF] Frisch | Investigating Multi-Touch and Pen Gestures for Diagram Editing on Interactive Surfaces[END_REF][START_REF] Isenberg | Interactive Exploratory Visualization of 2D Vector Fields[END_REF][START_REF] North | Understanding Multitouch Manipulation for Surface Computing[END_REF]) much remains to be learned about employing direct-touch input specifically for three-dimensional scientific visualization.

CONSTRAINTS OF SCIENTIFIC VISUALIZATION

Traditionally, PC-based environments or dedicated hardware setups [START_REF] Bowman | 3D User Interfaces: Theory and Practice[END_REF] such as virtual reality environments (e. g., the Responsive Workbench [START_REF] Krüger | The Responsive Workbench[END_REF] or the CAVE [START_REF] Bryson | Virtual Reality in Scientific Visualization[END_REF]) have often been employed for scientific visualization. While such settings have numerous advantages for creating and viewing visualizations, both have disadvantages for the interaction with 3D data. In PC settings, for instance, one typically interacts indirectly though a mouse. In VR environments one can interact directly by means of tracked objects in physical space (e. g., wands) but this type of control often leaves viewers with the feeling of interaction in "empty space" without adequate haptic feedback or rest positions. Touch interaction, in contrast, provides direct control (in 2D) with somesthetic feedback [START_REF] Robles-De-La-Torre | The Importance of the Sense of Touch in Virtual and Real Environments[END_REF] which can alleviate this problem by allowing users to feel "in control of the data." Unlike manipulating 2D data, however, touch interaction with 3D data or within 3D environments is a challenging task [START_REF] Steinicke | Multi-Touching 3D Data: Towards Direct Interaction in Stereoscopic Display Environments coupled with Mobile Devices[END_REF] because it requires an under-constrained mapping from 2D input parameters (touch) to transformations in 3D space (mouse-based interaction faces the same problem). To address this issue of 2D-to-3D mapping, several interaction techniques have been described in the past [START_REF] Coffey | Slice WIM: A Multi-Surface, Multi-Touch Interface for Overview+Detail Exploration of Volume Datasets in Virtual Reality[END_REF][START_REF] Cohé | tBox: A 3D Transformation Widget Designed for Touch-Screens[END_REF][START_REF] Edelmann | The DabR -A Multitouch System for Intuitive 3D Scene Navigation[END_REF][START_REF] Forlines | Adapting a Single-User, Single-Display Molecular Visualization Application for Use in a Multi-User, Multi-Display Environment[END_REF][START_REF] Fu | Multi-Touch Techniques for Exploring Large-Scale 3D Astrophysical Simulations[END_REF][START_REF] Hachet | Navidget for Easy 3D Camera Positioning from 2D Inputs[END_REF][START_REF] Hancock | Shallow-Depth 3D Interaction: Design and Evaluation of One-, Two-and Three-Touch Techniques[END_REF][START_REF] Hancock | Sticky Tools: Full 6DOF Force-Based Interaction for Multi-Touch Tables[END_REF][START_REF] Jung | Adapting X3D for Multi-Touch Environments[END_REF][START_REF] Martinet | 3D Positioning Techniques for Multi-Touch Displays[END_REF][START_REF] Martinet | Design and Evaluation of 3D Positioning Techniques for Multitouch Displays[END_REF][START_REF] Reisman | A Screen-Space Formulation for 2D and 3D Direct Manipulation[END_REF][START_REF] Wilson | Simulating Grasping Behavior on an Imaging Interactive Surface[END_REF][START_REF] Wilson | Bringing Physics to the Surface[END_REF][START_REF] Yu | FI3D: Direct-Touch Interaction for the Exploration of 3D Scientific Visualization Spaces[END_REF] and from which we can take inspiration. Yet, scientific visualization has a number of unique characteristics which make dedicated research necessary.

Only few of the previously named approaches [START_REF] Coffey | Slice WIM: A Multi-Surface, Multi-Touch Interface for Overview+Detail Exploration of Volume Datasets in Virtual Reality[END_REF][START_REF] Fu | Multi-Touch Techniques for Exploring Large-Scale 3D Astrophysical Simulations[END_REF][START_REF] Yu | FI3D: Direct-Touch Interaction for the Exploration of 3D Scientific Visualization Spaces[END_REF] deal with the specific needs and requirements of scientific visualization: First, scientific visualization has broad applicability and developed techniques cannot be specific to one type of digital object or a specific geometric projection. Virtually all of the above referenced methods address the 3D touch interaction problem only for a specific type of object (e. g., medium sized closed shape, planar representation, or particle cloud) in a specific environment (e. g., top-down projected space with ground plane or object(s) freely floating in space). Our goal is to develop and encompassing interaction landscape for scientific visualization which supports multiple different interaction methodologies in an environment that can change depending on the type of data being visualized. For example, when interacting with a volumetric dataset it is often necessary to independently control one or more cutting planes (i. e., planar objects) that are used to reveal the inside of the dataset together with an interaction of particle layer data (i. e., many points too small to use their surface to constrain an interaction). Secondly, for scientific visualization it is essential that interaction can be controlled precisely in 3D for space-accurate exploration. Many existing techniques lack the necessary precise control (e. g., [START_REF] Wilson | Bringing Physics to the Surface[END_REF]) for scientific exploration and even the ones with good control are still subject to the inherent imprecision of direct-touch input. Lastly, scientific visualization interaction includes many dedicated actions beyond general 3D navigation and object manipulation. These additional interactions are essential to scientific data exploration and include selection, object and parameter manipulation, interaction with the time axis, etc.

FUTURE RESEARCH DIRECTIONS

As a consequence of these unique constraints, we suggest a number of research directions for facilitating the exploratory interaction with three-dimensional scientific datasets using touch-sensitive displays.

Integrated Direct-Touch Interaction Toolkit

First, we plan to develop a toolkit for three-dimensiona directtouch interaction with scientific data. This toolkit needs to comprise a set of integrated techniques and methods to support two main visualization feedback loops: the data manipulation loop and the exploration and navigation loop [START_REF] Ware | Information Visualization -Perception for Design[END_REF]. The toolkit's purpose, therefore, is to make the 3D interaction techniques readily accessible for a variety of scientific visualization applications and also to outside researchers. The data manipulation loop for visualization application includes basic data interactions such as selection and positioning of objects in space. While these are basic operations, they are fundamental to many follow-up interactions in the navigation and exploration loop. For example, data representations need to be found, selected, and possibly positioned before they can be effectively compared or correlated. Interactions for the navigation and exploration loop are complex as they need to encompass theories of pathfinding, map use, spatial metaphors, awareness, and feedback [START_REF] Ware | Information Visualization -Perception for Design[END_REF].

The development of techniques dedicated to scientific visualization for both loops needs to start by developing data manipulation methods for general view changes (i. e., camera or projection manipulations) not only for the visualized data but also for dedicated data exploration elements such as cutting planes or drilling tools. Therefore, it is necessary to support a catalog of interaction needs for scientific visualizationabstracted across datasets and tasks. For this purpose it will be necessary to analyze several existing scientific visualization tools from various application domains and data types (e. g., brain visualization, astronomic particle visualization, fluid flow simulation) and identify their most fundamental and common interaction requirements. The resulting toolkit may not need to completely re-invent new 3D interaction techniques but may incorporate some of the previously developed approaches for direct-touch interaction with 3D objects [START_REF] Coffey | Slice WIM: A Multi-Surface, Multi-Touch Interface for Overview+Detail Exploration of Volume Datasets in Virtual Reality[END_REF][START_REF] Cohé | tBox: A 3D Transformation Widget Designed for Touch-Screens[END_REF][START_REF] Edelmann | The DabR -A Multitouch System for Intuitive 3D Scene Navigation[END_REF][START_REF] Forlines | Adapting a Single-User, Single-Display Molecular Visualization Application for Use in a Multi-User, Multi-Display Environment[END_REF][START_REF] Fu | Multi-Touch Techniques for Exploring Large-Scale 3D Astrophysical Simulations[END_REF][START_REF] Hachet | Navidget for Easy 3D Camera Positioning from 2D Inputs[END_REF][START_REF] Hancock | Shallow-Depth 3D Interaction: Design and Evaluation of One-, Two-and Three-Touch Techniques[END_REF][START_REF] Hancock | Sticky Tools: Full 6DOF Force-Based Interaction for Multi-Touch Tables[END_REF][START_REF] Jung | Adapting X3D for Multi-Touch Environments[END_REF][START_REF] Martinet | 3D Positioning Techniques for Multi-Touch Displays[END_REF][START_REF] Martinet | Design and Evaluation of 3D Positioning Techniques for Multitouch Displays[END_REF][START_REF] Reisman | A Screen-Space Formulation for 2D and 3D Direct Manipulation[END_REF][START_REF] Wilson | Simulating Grasping Behavior on an Imaging Interactive Surface[END_REF][START_REF] Wilson | Bringing Physics to the Surface[END_REF][START_REF] Yu | FI3D: Direct-Touch Interaction for the Exploration of 3D Scientific Visualization Spaces[END_REF]. However, care must be taken to make the different modes of manipulation compatible with each other.

In a second stage, it will be necessary to integrate methods for the exploration and navigation loop. For this purpose one first needs to add selection strategies that are compatible with the view selection techniques. The selection techniques also need to go beyond the common tap-to-select because scientific datasets can comprise a variety of different data types (e. g., volumetric data, particle data, line data, or surface data). Based on the ability to select data and/or subspaces, mechanisms for the manipulation of selected objects (relocate, reorient, or resize), for specification of parameters (e. g., transfer function manipulation, placing seed particles, etc.), for interaction with the scale of the displayed dataset (potentially across several magnitudes of scale), and many others need to be integrated. Moreover, domain-specific interactiontechniques need to be supported, e. g., specific ones for geological data [START_REF] Sultanum | Touching the Depths: Introducing Tabletop Interaction to Reservoir Engineering[END_REF][START_REF] Sultanum | Point it, Split it, Peel it, View it": Techniques for Interactive Reservoir Visualization on Tabletops[END_REF]. Similar to the constraint for the selection techniques, also the data manipulation techniques need to be compatible with the remaining techniques of the toolkit. One of the major challenges, therefore, will be to provide the set of interaction techniques in an integrated manner such that they do not negatively affect each other.

Precise Control Issues

An important additional challenge that arises when employing direct-touch interaction is that touch input is inherently imprecise due to the size of our fingers as interaction toolswhile scientific visualization often comes with a requirement of precise location and control of 3D data. Here, two aspects of precise control play an important role. The first aspect is the translation of imprecise touch input into control of similar precision as the mouse. Here, we can learn from HCI research which in the past has developed several strategies to provide such precision (e. g., [START_REF] Benko | Precise Selection Techniques for Multi-Touch Screens[END_REF]). The second aspect of precise control arises from scientific visualization's need to single out specific parameters and to control them without affecting others. This aspect implies that, in addition to fully integrated interactions, we need to also support partial interactions. For example, instead of only the known pinching interaction (RST, [START_REF] Hancock | Rotation and Translation Mechanisms for Tabletop Interaction[END_REF][START_REF] Reisman | A Screen-Space Formulation for 2D and 3D Direct Manipulation[END_REF]), scientific visualization needs support for navigation along or rotation around a single axis. Therefore, an Interaction Toolkit needs to support techniques that allow users to single-out certain parameters.

Stereoscopic Displays

Additional challenges arise when one wants to retain the benefits of direct-touch control but, at the same time, wants to take advantage of the improved depth perception provided by traditional dedicated visualization environments with stereoscopic displays. Research has shown that touch interaction with stereoscopic displays is challenging because it is strongly affected by parallax between the images displayed for both eyes [START_REF] Valkov | 2D Touching of 3D Stereoscopic Objects[END_REF]. Only when a virtual object is within a small distance from the screen can people perceive their touches as affecting the virtual objects [START_REF] Valkov | Touching Floating Objects in Projection-based Virtual Reality Environments[END_REF]. Moreover, when viewing an object displayed as reaching out of a display people often attempt to touch "in thin air." In contrast-if an object is displayed "below the surface"-people may not even perceive the display surface as being present and hit the (to them invisible) display in an attempt to touch the object behind it. While solutions such as transparent props [START_REF] Schmalstieg | Using Transparent Props for Interaction with the Virtual table[END_REF], separating the touch surface from the stereoscopic display [START_REF] Hachet | Toucheo: Multitouch and Stereo Combined in a Seamless Workspace[END_REF], or tilted setups in connection with shallow-depth data [START_REF] Butkiewicz | Exploratory Analysis of Ocean Flow Models with Stereoscopic Multi-Touch[END_REF] can be be used to alleviate this problem somewhat, these solutions lack support for dynamic visualization elements or the diverse character of scientific data in general.

Large Displays and Multi-User Settings

The large size of traditional visualization displays prompts the question of multi-user visualization settings, in particular if interaction techniques based on multi-touch input are used. Not only do certain 3D touch interaction techniques not scale well to large settings (e. g., [START_REF] Yu | FI3D: Direct-Touch Interaction for the Exploration of 3D Scientific Visualization Spaces[END_REF]) or are not compatible with vertical display setups (e. g., [START_REF] Wilson | Bringing Physics to the Surface[END_REF]), the possibility for several people using the same visualization environment simultaneously raises additional questions. For example, one could envision interactive discussions between colleagues, the use of touch-controlled visualization in group discussion, and the interactive touch-based presentation of visualizations to a larger audience. Yet, it is not simple to extend single-user interaction techniques to multi-user ones. For example, a single-user 3D exploration method being applied on a large wall display may not be suitable for a presentation setting since the interacting person is largely occluding the interaction and visualization space. In such situations it may be necessary to separate the interaction from the visualization space and to employ dedicated awareness techniques. An additional important challenge is that current multi-touch display technology does not support tracking of user identity. Without user identity one has to develop heuristics to determine which user issued a certain interface command and react to synchronous input accordingly. Multiple concurrent changes to an object require addressing computational challenges and, more importantly, how conflicts are handled.

CONCLUSION

We believe that-despite the discussed issues/challengestouch interaction can have a tremendous impact on how visualization is being used by domain scientists (and beyond). Direct-touch interaction has the potential to facilitate the use of scientific visualization on a much larger variety of display and user settings, instead of being restricted to largely singleuser, mouse/keyboard-based interaction in PC environments or specialized 3D visualization hardware. Thus, instead of only being the end product of a scientific exploration process, intuitive touch-based interactive visualization technology can be tightly integrated into the scientific exploration process, and could actively be used for gaining an understanding of the analyzed data. This means that scientists may be able to use 3D interaction techniques to not only discuss ideas but instead to collaboratively create and manipu-late visuals that illustrate their data, resulting in fundamental insights and an easy way to communicate these to others.


Gabriel Rivière

REMARKS ON QUANTUM ERGODICITY

Introduction

Let M be a smooth, compact, connected Riemannian manifold of dimension d (without boundary). Denote by L the normalized Liouville measure on the unit cotangent bundle S * M and by g t the geodesic flow on S * M . Let (ψ j ) j∈N be an orthonormal basis of eigenfunctions of -∆ g associated to a nondecreasing sequence of eigenvalues (λ 2 j ) j∈N , i.e.

-∆ g ψ j = λ 2 j ψ j , ψ j L 2 (M) = 1.

In the following, we will write N (λ) := ♯{j : λ 2 j ≤ λ 2 }. Our goal in this note is to describe the asymptotic distribution of this sequence of eigenfunctions as λ 2 j tends to infinity. For that purpose, we introduce the following distribution on S * M : ∀a ∈ C ∞ (S * M ), µ j (a) = S * M adµ j := ψ j , Op(a)ψ j , where Op(a) is a pseudodifferential operator with principal symbol a. It is a classical fact to check that any accumulation point 1 of this sequence belongs to the set M(S * M, g t ) of (g t ) t -invariant probability measures on S * M [START_REF] Gérard | Microlocal Defect Measures[END_REF][START_REF]Burq Mesures semi-classiques et mesures de défaut[END_REF][START_REF] Zworski | Semiclassical analysis[END_REF].

If the Liouville measure is ergodic, the Quantum Ergodicity Theorem states that there exists a subset S of density 2 1 in N such that the sequence (µ j ) j∈S converges to the Liouville measure L [START_REF]Shnirelman Ergodic properties of eigenfunctions[END_REF][START_REF]Zelditch Uniform distribution of the eigenfunctions on compact hyperbolic surfaces[END_REF][START_REF] De | Verdière Ergodicité et fonctions propres du Laplacien[END_REF]. In other words, it means that the eigenfunctions become equidistributed in S * M . We refer the reader to [START_REF]Zelditch Recent developments in mathematical quantum chaos[END_REF] for a recent detailed survey on related issues. In our context, the main example of application is given by geodesic flows on manifolds of negative curvature or more generally by uniformly hyperbolic geodesic flows: in this setting, the Liouville measure is known to be ergodic.

Here, we are interested in the case where we drop the ergodicity assumption.

The main examples we have in mind are geodesic flows for which the Liouville measure is ergodic on a subset of positive measure. For instance, this kind of situations occurs when the geodesic flow is supposed to be nonuniformly hyperbolic [START_REF] Barreira | Pesin Smooth ergodic theory and nonuniformly hyperbolic dynamics[END_REF]. In these cases, we derive properties on the asymptotic distributions of the eigenmodes. For that purpose, we prove an alternative version of the Quantum Ergodicity Theorem that does not rely on ergodicity. Then, we apply this result in several geometric contexts. For example, we obtain a kind of equidistribution property for subsequences of eigenfunctions of the Laplacian on surfaces of nonpositive curvature with genus ≥ 2. In this setting, the Liouville measure is not known to be ergodic; thus, the standard Quantum Ergodicity Theorem does not apply a priori.

Statement of the main result

Let Λ be a subset of S * M . We say that it satisfies the Birkhoff property if [START_REF] Barreira | Pesin Smooth ergodic theory and nonuniformly hyperbolic dynamics[END_REF] ∀ρ ∈ Λ, ∃ L ρ ∈ M(S * M, g t ), lim

T →+∞ 1 T T 0 δ g t ρ dt = L ρ ,
where the convergence is for the weak-⋆ topology, i.e.

∀a ∈ C 0 (S * M ), ∀ρ ∈ Λ, lim

T →+∞ 1 T T 0 a(g t ρ)dt = L ρ (a).
For every subset Λ ⊂ S * M satisfying property (1), we introduce Cv(Λ) which is the closure in D ′ (S * M ) of the convex hull of {L ρ : ρ ∈ Λ}. We emphasize that the set Cv(Λ) depends on the choice of Λ and that any element in Cv(Λ) is a probability measure on S * M invariant under the geodesic flow. Thanks to the Birkhoff Ergodic Theorem [6] -section 6.1, there exists Λ ⊂ S * M satisfying (1) and L(Λ) = 1. The main result of this note is the following version of the Quantum Ergodicity Theorem:

Theorem 2.1. Let Λ ⊂ S * M satisfying property (1) and such that L(Λ) = 1. Let (ψ j ) j∈N be an orthonormal basis of eigenfunctions of -∆ g associated to a nondecreasing sequence of eigenvalues (λ 2 j ) j∈N , i.e.

-∆ g ψ j = λ 2 j ψ j , ψ j L 2 (M) = 1. Then, there exists S ⊂ N of density 1 such that any accumulation point of the sequence (µ j ) j∈S belongs to Cv(Λ).

Remark 2.2. We emphasize that the result holds for any choice of Λ ⊂ S * M of full measure satisfying the Birkhoff property (1) (the result is of course more interesting when the set Cv(Λ) is not too big). We also underline that this theorem is true for any orthonormal basis of eigenfunctions of ∆ g and that we do not need to make any particular assumption on the manifold (like ergodicity for instance). Even if this generalization is quite natural, we did not find any trace of such a result in the literature.

If M is the sphere S d endowed with its canonical metric, the set Cv(Λ) is always equal to M(S * M, g t ) for Λ of full measure. Thus, the result is empty as we already know that any accumulation point of the sequence (µ j ) j≥0 is an invariant probability measure. In the "opposite" case where the Liouville measure is ergodic for the geodesic flow, we recover the standard Quantum Ergodicity Theorem [START_REF]Shnirelman Ergodic properties of eigenfunctions[END_REF][START_REF]Zelditch Uniform distribution of the eigenfunctions on compact hyperbolic surfaces[END_REF][START_REF] De | Verdière Ergodicité et fonctions propres du Laplacien[END_REF] as, thanks to the Birkhoff Ergodic Theorem, one can pick Λ of full measure satisfying [START_REF] Barreira | Pesin Smooth ergodic theory and nonuniformly hyperbolic dynamics[END_REF] and L ρ = L on Λ (and thus Cv(Λ) = {L}). In section 3, we will provide other examples of applications.

In the physics literature, the "semiclassical eigenfunctions hypothesis" states that the eigenmodes (ψ j ) j≥0 must be asymptotically concentrated into regions of phase space which a typical orbit explores in the long time limit [START_REF]Regular and irregular spectra[END_REF][START_REF] Berry | Regular and irregular semiclassical wavefunctions[END_REF]. In our context, the set Λ could represent in some sense a set of typical orbits and the measure L ρ is the canonical measure associated to the orbit of a point ρ in the phase space S * M .

Regarding this conjecture, it seems natural to understand when there exist a subset Λ of full measure and a typical subsequence of eigenmodes (ψ j ) j∈S such that the accumulation points of (µ j ) j∈S are exactly given by the closure of {L ρ : ρ ∈ Λ}. This question was for instance raised by Shnirelman in [START_REF] Shnirelman | Asymptotic properties of eigenfunctions in the regions of chaotic motion, Addendum to KAM theory and semiclassical approximations of eigenfunctions[END_REF] -end of paragraph AD.2. In such generality, it is not true as there exist geometric situations where the set {L ρ : ρ ∈ Λ} cannot be reduced to L while there exists a typical family of states that converges to L [START_REF] Zelditch | Quantum ergodicity on the sphere[END_REF][START_REF]Zelditch Recent developments in mathematical quantum chaos[END_REF][START_REF] Gutkin | Note on converse quantum ergodicity[END_REF][START_REF]Zelditch Random orthonormal bases of spaces of high dimension[END_REF].

Our theorem shows that, for a typical family (µ j ) j∈S , the accumulation points belong to a larger set than the closure of {L ρ : ρ ∈ Λ}, precisely they belong to the closure of its convex hull. We underline that results related to these questions were also obtained by Marklof and O'Keefe for specific families of quantum maps with divided phase space [START_REF] Marklof | Weyl's law and quantum ergodicity for maps with divided phase space, with an appendix by S. Zelditch[END_REF], and by Galkowski for Hamiltonian flows with divided phase space on manifolds with piecewise smooth boundary [START_REF] Galkowski | Quantum Ergodicity for a Class of Mixed Systems[END_REF].

We will explain in paragraph 4 how one can get our generalized version of the Quantum Ergodicity Theorem by implementing an idea used by Sjöstrand in the context of damped wave equations [START_REF] Sjöstrand | Asymptotic distribution of egenfrequencies for damped wave equations[END_REF]. In fact, our proof will combine Hahn-Banach Theorem with the following main lemma that makes the connection with the results in [START_REF] Sjöstrand | Asymptotic distribution of egenfrequencies for damped wave equations[END_REF] more explicit: As in the classical proof of the Quantum Ergodicity Theorem, the main ingredient of the proof of this lemma is a local Weyl law combined to an averaging of a along geodesics and to a positivity argument.

Remark 2.4. By construction, ρ → L ρ (a) belongs to L ∞ (S * M, dL), and thus| essinf L ρ (a)|, | esssup L ρ (a)| < +∞. Compared with our main theorem, these two quantities do not depend on any choice of subset Λ. For any subset Λ of full measure satisfying (1), one has

inf ρ∈Λ L ρ (a) ≤ essinf L ρ (a) ≤ esssup L ρ (a) ≤ sup ρ∈Λ L ρ (a).
Organization of the following. In the next paragraph, we apply our theorem in several geometric situations. After that, we give the proof of theorem 2.1 and provide the proof of an intermediary proposition that we used in our applications.

In the following, we will sometimes use the notation D(S * M ) for the space C ∞ (S * M ) when we want to emphasize that we are working with distributions.

Application of theorem 2.1

Before entering the details of the proof, we describe several geometric situations where the Liouville measure is a priori not supposed to be ergodic and we apply theorem 2.1 in order to derive some weak equidistribution properties for the eigenfunctions.

3.1. Geodesic flows with divided phase space. A direct consequence of theorem 2.1 is the following property: Corollary 3.1. Suppose there exist Λ ′ of positive Liouville measure, I finite and a family (Λ i ) i∈I of invariant subsets such that

• L(Λ i ) > 0 for every i ∈ I; • Λ i ∩ Λ j is empty when i = j; • ∪ i Λ i = Λ ′ ; • L |Λi is ergodic for every i ∈ I.
Then, for every Λ 0 ⊂ Λ ′c satisfying (1) and L(Λ 0 ) = L(Λ ′c ), there exists a subset S ⊂ N of density 1 such that any accumulation point of the sequence (µ j ) j∈S is of the form

(2) µ = α i∈I t i L |Λi L(Λ i ) + (1 -α)ν 0 , where 0 ≤ α ≤ 1, 0 ≤ t i ≤ 1, i∈I t i = 1 and ν 0 belongs to Cv(Λ 0 ).
In the case where L(Λ ′ ) = 1, this corollary tells us that, along a subsequence S of density 1, any accumulation point of the sequence (µ j ) j∈S must be absolutely continuous with respect to L. Moreover, it gives us an explicit expression for the accumulation points. Galkowski recently proved a similar result in the case of manifolds with piecewise smooth boundaries under the assumption that Λ ′ satisfies L |Λ ′ ergodic, L(Λ ′ ) > 0 and L(∂Λ ′ -Λ ′ ) = 0 [START_REF] Galkowski | Quantum Ergodicity for a Class of Mixed Systems[END_REF]. It would be interesting to understand if theorem 2.1 could also be obtained in the setting of manifolds with piecewise smooth boundaries.

Remark 3.2. Even if there is no hypothesis on nonuniform hyperbolicity in the above statement, we underline that the assumptions of the corollary appear naturally in this context -we refer the reader to [START_REF] Barreira | Pesin Smooth ergodic theory and nonuniformly hyperbolic dynamics[END_REF] for recent lecture notes on this topic. Using Pesin classical work on nonuniform hyperbolicity [START_REF] Pesin | Characteristic Ljapunov exponents, and smooth ergodic theory[END_REF] (see also Theorem 11.5 in [START_REF] Barreira | Pesin Smooth ergodic theory and nonuniformly hyperbolic dynamics[END_REF]), we recall that, if there exists an invariant subset Λ ⊂ S * M of positive Liouville measure such that the geodesic flow is nonuniformly hyperbolic on Λ, then there exists I at most countable and a family (Λ i ) i∈I of invariant subsets such that

• ∪ i∈I Λ i = Λ (modulo a subset of 0 Liouville measure); • L(Λ i ) > 0 for every i ∈ I; • Λ i ∩ Λ j is empty when i = j; • L |Λi is ergodic for every i ∈ I.
Remark 3.3. In [START_REF] Gutkin | Note on converse quantum ergodicity[END_REF], Gutkin constructs a nonuniformly hyperbolic geodesic flow on a billiard table for which the Liouville measure is not ergodic. However, his system presents a symmetry that allows him to obtain a stronger result than corollary 3.1, precisely he proves the existence of a subsequence of density 1 converging to the Liouville measure. It is not clear to the author whether there exist smooth compact Riemannian manifolds without boundary which satisfy the assumption of the corollary with L(Λ ′ ) = 1 and |I| ≥ 2, and which do not present a symmetry like the one in [START_REF] Gutkin | Note on converse quantum ergodicity[END_REF]. We also do not know if ergodicity or even I finite are "generic properties" in the family of smooth nonuniformly hyperbolic geodesic flows.

Remark 3.4. In order to provide a concrete example, we briefly discuss a construction due to Donnay [START_REF] Donnay | Geodesic flow on the two-sphere. I. Positive measure entropy[END_REF] -section 11. In this reference, he constructs Riemmanian metrics on the sphere S 2 for which the phase space splits into a chaotic component and a regular one. His strategy is to remove three or more points from the sphere and to endow the induced punctured surface with the Poincaré metric; then, he attaches smoothly a so called "light-bulb cap" in a neighborhood of each deleted point. The geodesic flow he obtains is not ergodic for L. The phase space S * M contains two disjoint invariant subsets Λ chaotic and Λ regular of positive Liouville measure satisfying L(Λ chaotic ∪ Λ regular ) = 1. More precisely, Λ regular consists of orbits that stay in the caps and g t |Λ chaotic is nonuniformly hyperbolic. Thanks to remark 3.2, the corollary applies here. 3.2. More on divided phase space. The statements of the previous paragraph do not forbid that the ergodic component associated to a subset Λ i of positive Liouville measure has a weight t i = 0. This motivates the following proposition which gives partial informations on this question. Proposition 3.5. Suppose there exists an invariant subset Λ ′ such that Λ ′ contains a nonempty open ball (modulo a subset of 0 Liouville measure) and such that L |Λ ′ is ergodic. Then, there exists Λ 1 ⊂ Λ ′c satisfying L(Λ 1 ) = L(Λ ′c ) such that, for every Λ 0 ⊂ Λ 1 verifying (1) and L(Λ 1 ) = L(Λ 0 ) and for every 0 ≤ δ < L(Λ ′ ), there exists S δ ⊂ N of density 3 ≥ δ such that any accumulation point of the sequence (µ j ) j∈S δ is of the form (2) with

0 < L(Λ ′ ) -δ 1 -δ ≤ α ≤ 1.
Even if the conclusion of the proposition is slightly technical, we emphasize that we only need to assume that Λ ′ contains a nonempty open ball (modulo a subset of 0 Liouville measure) in order to have positive mass on the ergodic component Λ ′ .

This result is true for any orthonormal basis of eigenfunctions of ∆ g . It can be obtained as an application of lemma 2.3 -see paragraph 5 for details. It provides a sufficient condition to observe a kind of equidistribution property on an invariant subset Λ ′ -see [START_REF] Shnirelman | Asymptotic properties of eigenfunctions in the regions of chaotic motion, Addendum to KAM theory and semiclassical approximations of eigenfunctions[END_REF] for related questions.

Remark 3.6. The hypothesis is slightly different from the one of corollary 3.1 but the argument could be adapted to treat the case where the assumptions of the corollary are satisfied with Λ ′ containing a nonempty open ball (modulo a subset of 0 Liouville measure) -see the end of paragraph 5 for details. Precisely, we will check that there exists Λ 1 ⊂ Λ ′c satisfying L(Λ 1 ) = L(Λ ′c ) such that, for every Λ 0 ⊂ Λ 1 verifying (1) and L(Λ 1 ) = L(Λ 0 ), there exists a subset S ′ of 3 By density ≥ δ, we mean that lim inf

λ→+∞ 1 N (λ) ♯{j : λ 2 j ≤ λ 2 and j ∈ S δ } ≥ δ.
positive density such that any accumulation point of the sequence (µ j ) j∈S ′ is of the form given by equation ( 2) with α > 0.

3.3. Surfaces of nonpositive curvature. We will now give an application of the previous proposition in the context of nonpositively curved manifolds. We suppose that M is a surface of nonpositive curvature of genus ≥ 2. For any point x in M , we will denote by K(x) ≤ 0 the sectional curvature at point x. We will make a small abuse of notations and use also the notation K for its canonical lift on S * M . Following [START_REF] Barreira | Pesin Smooth ergodic theory and nonuniformly hyperbolic dynamics[END_REF], we introduce the following invariant subset

(3)

Λ ′ := ρ ∈ S * M : lim sup T →+∞ 1 T T 0 K • g t (ρ)dt < 0 .
According to [START_REF] Barreira | Pesin Smooth ergodic theory and nonuniformly hyperbolic dynamics[END_REF] (Theorems 2.4 and 17.7), the set Λ ′ is open (modulo a subset of 0 Liouville measure) and it satisfies L(Λ ′ ) > 0 and L |Λ ′ is ergodic. In particular, the subset Λ ′ satisfies all the requirements of proposition 3.5. It is also shown in this reference that Λ ′ is everywhere dense but we will not use this fact here.

We underline that it is still an open question to determine whether L(Λ ′ ) = 1 or not for any surface of nonpositive curvature of genus ≥ 2. In other words, it is not known if the Liouville measure is ergodic or not. Yet, we can prove that the eigenfunctions satisfy some equidistribution properties in this negatively curved part of the surface. For that purpose, we observe that, thanks to Birkhoff Ergodic Theorem, L ρ (K) is well defined a.e. on Λ ′c . By definition of Λ ′ , we obtain

L ρ (K) = 0 a.e. on Λ ′c .
As the subset Λ ′ satisfies all the requirements of proposition 3.5, the following corollary holds:

Corollary 3.7. If M is a surface of nonpositive curvature K(x) and of genus ≥ 2, then there exists Λ 1 ⊂ {L ρ (K) = 0} ∩ Λ ′c (with L(Λ 1 ) = L(Λ ′c )) such that, for any Λ 0 ⊂ Λ 1 satisfying (1)
and L(Λ 0 ) = L(Λ ′c ), and, for every 0 ≤ δ < L(Λ ′ ),

(1) there exists a subset S of density 1 in N such that any accumulation point of the sequence (µ j ) j∈S is of the form

α L |Λ ′ L(Λ ′ ) + (1 -α)ν 0 ,
where 0 ≤ α ≤ 1 and ν 0 belongs to Cv(Λ 0 ); (2) there exists S δ ⊂ S of density ≥ δ such that, for any accumulation point of the sequence (µ j ) j∈S δ , one has

0 < L(Λ ′ ) -δ 1 -δ ≤ α ≤ 1.
This result is true for any orthonormal basis of eigenfunctions of ∆ g . This corollary tells us that a positive proportion of eigenmodes are asymptotically equidistributed in Λ ′ even if we do not have ergodicity of the Liouville measure on the entire phase space. We emphasize that the assumptions on Λ 0 implies that ν 0 (K) = 0 for any ν 0 in Cv(Λ 0 ). Remark 3.8. If we project the distributions µ j on the base, we find the following notable consequence of the previous corollary: any accumulation point of the sequence (K|ψ j | 2 vol M ) j∈S is of the form cKvol M where c ≥ 0 is a constant. Moreover, thanks to part (2) of the corollary, there exist subsequences of positive density for which c can be chosen positive. Remark 3.9. In the case where dim M ≥ 2, one can introduce the following subset of S * M :

Λ ′ := ρ ∈ S * M : lim sup T →+∞ 1 T T 0 K π•g t ρ (g t ρ, g t ρ ′ )dt < 0, for every ρ ′ orthogonal to ρ ,
where π : S * M → M is the canonical projection on M and K x (v 1 , v 2 ) is the sectional curvature for x in M and v 1 , v 2 in T *

x M . Suppose now that M has nonpositive curvature, i.e. K x (v 1 , v 2 ) ≤ 0 for every x in M and every v 1 , v 2 in T *

x M . Under some extra geometric assumptions 4 on M that are always satisfied by nonpositively curved surfaces of genus ≥ 2, the set Λ ′ is again open (modulo 0) and everywhere dense and it satisfies L(Λ ′ ) > 0 and L |Λ ′ is ergodic. Then, corollary 3.7 can be extended in dim M ≥ 2 modulo the above extra geometric assumptions.

3.4. Flat torus. In this paragraph, we apply theorem 2.1 to the flat torus T d for which there is no subset Λ ′ of positive measure on which L is ergodic. We introduce the subset of "irrational" vectors

Λ := (x, ξ) ∈ S * T d : ∀p ∈ Z d -{0}, p.ξ = 0 .
This set has full Liouville measure and Λ satisfies property (1) with L ρ = dx × δ ξ . In particular, the projection on T d of any element in Cv(Λ) is the Lebesgue measure dx. Applying theorem 2.1, we obtain the following corollary:

Corollary 3.10. For any orthonormal basis (ψ j ) j∈N of eigenfunctions of ∆ on T d , there exists a subset S of density 1 in N such that

∀a ∈ C 0 (T d ), lim j→+∞,j∈S T d a(x)|ψ j (x)| 2 dx = T d a(x)dx.
This result is the analogue on T d of Marklof-Rudnick's recent result on equidistribution of eigenfunctions on rational polygons [START_REF] Marklof | Rudnick Almost all eigenfunctions of a rational polygon are uniformly distributed[END_REF].

Proof of the main result

The proof follows classical ideas taken from [START_REF]Shnirelman Ergodic properties of eigenfunctions[END_REF][START_REF]Zelditch Uniform distribution of the eigenfunctions on compact hyperbolic surfaces[END_REF][START_REF] De | Verdière Ergodicité et fonctions propres du Laplacien[END_REF][START_REF] Sjöstrand | Asymptotic distribution of egenfrequencies for damped wave equations[END_REF] that we carefully combine to get our main theorem. 4.1. Preliminary remarks. Before getting into the details of the proof, we mention a few facts on the properties of the quantization procedure Op. First, recall that it is not defined in a canonical way: it depends on a choice of coordinate charts and on a choice of quantization procedure on R 2d -see section II.5 in [START_REF] Taylor | Pseudodifferential operators[END_REF] or chapter 14 in [START_REF] Zworski | Semiclassical analysis[END_REF]. A standard choice is to take the Weyl quantization on R 2d . Then, for two different choices of coordinate charts, we obtain two quantization procedures Op and Op ′ . Yet, one can show that, for any a in C ∞ (S * M ), Op(a)-Op ′ (a) is a pseudodifferential operator of order -1. In particular, one has lim j→+∞ ψ j , (Op(a) -Op ′ (a))ψ j L 2 (M) = 0, and thus the accumulation points of the sequence (µ j ) j≥0 do not depend on the choice of coordinates on M . Let us now briefly recall a few properties of this quantization procedure that we will use at different steps of our argument.

For every a in C ∞ (S * M ), Op(a) defines a bounded operator in L 2 (M ). Moreover, if a is real valued, then Op(a) is selfadjoint. In particular, the sequence of distributions µ j is real valued, i.e. µ j (a) belongs to R when a is real valued.

We introduce the average of a ∈ C ∞ (S * M ) at time T > 0,

a T (ρ) := 1 T T 0 a • g t (ρ)dt.
As (ψ j ) j≥0 is a sequence of eigenmodes of ∆ g , one can apply the Egorov Theorem (theorem 15.2 in [START_REF] Zworski | Semiclassical analysis[END_REF]) and get

(4) ∀T > 0, µ j (a) = µ j (a T ) + o T (1),
where o T (1) is a remainder that depends on a and T and that tends to 0 as j tends to infinity. We will also use the following local Weyl law (theorem 15.3 in [START_REF] Zworski | Semiclassical analysis[END_REF]):

(

5) ∀b ∈ C ∞ (S * M ), 1 N (λ) j:λ 2 j ≤λ 2 µ j (b) = S * M bdL + o(1), as λ → +∞,
where the remainder depends on b.

An inconvenience of choosing the Weyl quantization on R 2d is that it does not satisfy a positivity property. This can be solved by using a different quantization on R 2d . For instance, as in the proof of [START_REF] De | Verdière Ergodicité et fonctions propres du Laplacien[END_REF] (section 1.1), one can take the so-called Friedrichs quantization on R 2d -see section V II.2 in [START_REF] Taylor | Pseudodifferential operators[END_REF] for details. After choosing a family of coordinate charts, it gives us a quantization procedure

Op + on M such that, for every b ∈ C ∞ (S * M ), b ≥ 0 =⇒ Op + (b) ≥ 0.
The Weyl and the Friedrichs quantization are "equivalent" on R 2d -Theorem 2.2 of Ch. V II [START_REF] Taylor | Pseudodifferential operators[END_REF]. In particular, one has, for every b ∈ C ∞ (S * M ), [START_REF] Einsiedler | Ward Ergodic Theory: with a view towards Number Theory[END_REF] lim

j→+∞ ψ j , (Op(b) -Op + (b))ψ j L 2 (M) = 0.
Thus, the accumulation points of the sequence µ j do not change if we replace Op by Op + in the definition of µ j . Moreover, the invariance relation ( 4) and the local Weyl law ( 5) are still valid if we replace Op by Op + in the definition of µ j .

4.2.

Proof of lemma 2.3. We start our proof by giving the proof of the main lemma 2.3. Let a be an element in C ∞ (S * M, R). In order to simplify the presentation, denote A 0 := esssup L ρ (a)see remark 2.4. By definition, one has that, for every δ > 0,

(7) L ({ρ ∈ S * M : a T (ρ) ≥ A 0 + δ}) → 0, as T → +∞.
Fix now T > 0 and ǫ > 0. Properties ( 4) and ( 6) tell us that, as j tends to ∞,

µ j (a) = µ j (a T ) + o T (1) = ψ j , Op + (a T )ψ j + o T (1),
where the remainder depends on T . As in [START_REF] Sjöstrand | Asymptotic distribution of egenfrequencies for damped wave equations[END_REF], one can define a new smooth function ãT ≤ a T on

S * M such that • ãT (ρ) = a T (ρ) when a T (ρ) ≤ A 0 + √ ǫ 2 . • ãT (ρ) ≤ A 0 + √ ǫ otherwise.
Remark 4.1. In order to construct the function ãT , one can fix an increasing smooth function

θ ǫ : R → R which satisties ∀v ∈ R, θ ǫ (v) ≤ v, |θ ′ (v)| ≤ 1,
and

∀v ≤ A 0 + √ ǫ 2 , θ ǫ (v) = v, and θ ǫ (v) ≤ A 0 + √ ǫ otherwise.
Then, one defines ãT = θ ǫ (a T ). In particular, ãTa T ∞ is bounded independently of T > 0.

As j tends to infinity, one has the following equality:

µ j (a) = ψ j , Op + (a T -ãT )ψ j + ψ j , Op + (ã T )ψ j + o T (1).
The idea of introducing this new function is taken from [START_REF] Sjöstrand | Asymptotic distribution of egenfrequencies for damped wave equations[END_REF] where it was used to study spectral asymptotics of the damped wave equation. In the following lines, we will show that

• most of the terms in the sequence ( ψ j , Op + (a T -ã T )ψ j ) j≥0 are small following arguments from [START_REF]Zelditch Uniform distribution of the eigenfunctions on compact hyperbolic surfaces[END_REF][START_REF] De | Verdière Ergodicité et fonctions propres du Laplacien[END_REF]; • the other term in the RHS will be less than A 0 + √ ǫ + o T (1) by construction. A careful combination of these two facts will finally allow us to get our conclusion.

From [START_REF] Einsiedler | Ward Ergodic Theory: with a view towards Number Theory[END_REF] and the local Weyl law (5), one has

1 N (λ) λ 2 j ≤λ 2 ψ j , Op + (a T -ãT )ψ j = S * M (a T -ãT )dL + o T (1),
where each term in the sum is nonnegative (as a T -ãT ≥ 0). Thanks to our construction, one has

S * M (a T -ãT )dL ≤ C a,ǫ L ρ ∈ S * M : a T (ρ) ≥ A 0 + √ ǫ 2 .
Fix η > 0. Combining [START_REF] Galkowski | Quantum Ergodicity for a Class of Mixed Systems[END_REF] to the two previous relations, there exists T ǫ,η > 0 such that, for every T ≥ T ǫ,η , one can find λ T > 0 satisfying

λ ≥ λ T =⇒ 1 N (λ) λ 2 j ≤λ 2 ψ j , Op + (a T -ãT )ψ j ≤ ηǫ.
We now fix T = T ǫ,η . Denote D ǫ (λ) := {j : λ 2 j ≤ λ 2 and ψ j , Op + (a T -ãT )ψ j ≥ √ ǫ}. Thanks to the Tchebychev inequality, we obtain that ♯Dǫ(λ) N (λ) ≤ η √ ǫ for λ ≥ λ T . This means that most of the terms in the sequence of nonnegative numbers ( ψ j , Op + (a T -ãT )ψ j ) j≥0 are small.

As Op + is positive and ãT ≤ A 0 + √ ǫ, one has that, for λ 2 j larger than some A > 0, the term

ψ j , Op + (ã T )ψ j + o T (1) is less than A 0 + 2 √ ǫ.
Thanks to the above discusion, we can write

♯ j : λ 2 j ≤ λ 2 and µ j (a) ≤ A 0 + 3 √ ǫ ≥ ♯ j : λ 2 j ≤ A and µ j (a) ≤ A 0 + 3 √ ǫ +♯ j : A ≤ λ 2 j ≤ λ 2 and ψ j , Op + (a T -ãT )ψ j < √ ǫ . If we denote S ǫ := {j : µ j (a) ≤ A 0 + 3 √ ǫ}, then we have lim inf λ→+∞ ♯ S ǫ ∩ j : λ 2 j ≤ λ 2 N (λ) ≥ 1 -η √ ǫ.
This is true for any η > 0 which implies that S ǫ has density 1.

Using the procedure of paragraph 5 in [START_REF] De | Verdière Ergodicité et fonctions propres du Laplacien[END_REF] -see remark 4.2 below, one can then obtain a subset S 0 ⊂ N of density 1, such that any accumulation point of the sequence (µ j (a)) j∈S0 is ≤ A 0 . This achieves the proof of the upper bound in lemma 2.3 and the lower bound can be easily derived by considering -a. Remark 4.2. For the sake of completeness, we reproduce the argument used in [START_REF] De | Verdière Ergodicité et fonctions propres du Laplacien[END_REF] in order to construct the subset S 0 . Observe that the family of subsets (S 1 l ) l≥1 we have just defined is nonincreasing. We fix, for every l ≥ 2,

α 1 such that, ∀λ 2 ≥ α 1 , ♯ S 1 ∩ j : λ 2 j ≤ λ 2 N (λ) ≥ 1 - 1 2 , . . . , α l ≥ α l-1 such that, ∀λ 2 ≥ α l , ♯ S 1 l ∩ j : λ 2 j ≤ λ 2 N (λ) ≥ 1 - 1 2 l .
Then, one defines S 0 such that S 0 ∩ {j : α l ≤ λ 2 j < α l+1 } = S 1 l ∩ {j : α l ≤ λ 2 j < α l+1 }. In particular, for every l ≥ 1 and for every α l ≤ λ 2 < α l+1 , one has S 0 ∩ {j : 0 ≤ λ 2 j ≤ λ 2 } ⊃ S 1 l ∩ {j : 0 ≤ λ 2 j ≤ λ 2 } and thus, S 0 has density 1. By construction, any accumulation point of the sequence (µ j (a)) j∈S0 is ≤ A 0 . 4.3. Proof of theorem 2.1. We are now in position to prove theorem 2.1. For that purpose, we interpret lemma 2.3 as an inequality on linear forms and then we apply Hahn-Banach Theorem.

First, we observe that

inf ρ∈Λ L ρ (a) ≤ essinf L ρ (a) ≤ esssup L ρ (a) ≤ sup ρ∈Λ L ρ (a).
Fix now (a k ) k∈N a family of smooth functions which is dense in C 0 (S * M, R) (for the uniform topology). For every k ∈ N, lemma 2.3 gives us a subset S k of density 1. Without loss of generality 5 , one can suppose that (S k ) k∈N is a nonincreasing sequence of subsets. Using remark 4.2, one can choose a subset S of density 1 such that

∀k ∈ N, inf ρ∈Λ L ρ (a k ) ≤ lim inf j→+∞,j∈S µ j (a k ) ≤ lim sup j→+∞,j∈S µ j (a k ) ≤ sup ρ∈Λ L ρ (a k ).
Fix now an accumulation point µ of the sequence (µ j ) j∈S . By a density argument, the above inequality implies then

∀a ∈ D(S * M, R), inf ρ∈Λ L ρ (a) ≤ µ(a) ≤ sup ρ∈Λ L ρ (a)
. 5 The intersection of two subsets of density 1 is still a subset of density 1.

As the space D(S * M, R) is the topological dual of D ′ (S * M, R) (Theorem XIV , Chapter 3 in [START_REF] Schwartz | Théorie des distributions[END_REF]), the previous inequality implies that, for every continuous linear form Φ on D ′ (S * M, R),

inf ρ∈Λ Φ(L ρ ) ≤ Φ(µ) ≤ sup ρ∈Λ Φ(L ρ ).
Suppose by contradiction that µ does not belong to Cv(Λ). By the Hahn-Banach theorem [START_REF] Trèves | Topological vector spaces, distributions and kernels[END_REF], there exists a continuous linear form Φ 0 on D ′ (S * M, R) that strictly separates the closed convex subset Cv(Λ) from the compact convex subset {µ}. In particular, there exists α in R such that ∀ν ∈ Cv(L), Φ 0 (ν) ≤ α < Φ 0 (µ).

We get that sup ν∈Cv(L) Φ 0 (ν) ≤ α < Φ 0 (µ). In particular, sup ρ∈Λ Φ 0 (L ρ ) ≤ α < Φ 0 (µ) which leads to the contradiction.

Proof of proposition 3.5

In this final section, we will prove proposition 3.5 that we used in our applications to surfaces of nonpositive curvature.

As in the statement of the proposition, we fix an invariant subset Λ ′ in S * M containing a nonempty open ball (modulo a subset of zero Liouville measure) and satisfying L |Λ ′ ergodic. Let Λ 0 ⊂ Λ ′c satisfying L(Λ 0 ) = L(Λ ′c ). Thanks to corollary 3.1, we know that there exists S ⊂ N of density 1 such that any accumulation point of the sequence (µ j ) j∈S is of the form

α L |Λ ′ L(Λ ′ ) + (1 -α)ν 0 ,
where α ≥ 0 and where ν 0 belongs to Cv(Λ 0 ).

As Λ ′ contains a nonempty open ball (modulo a set of zero Liouville measure), one can pick χ ≥ 0 a smooth function which is compactly supported in this open ball and which is not equal to 0 everywhere. In particular, S * M χdL = Λ ′ χdL > 0.

Preliminary remarks. Recall from Birkhoff Ergodic Theorem that

L ρ (χ) = lim T →+∞ 1 T T 0 χ • g t (ρ)dt
is well defined for L almost every ρ in S * M . From our assumptions on Λ ′ , one can verify that

L ρ (χ) = 1 L(Λ ′ ) Λ ′ χdL, a.e. on Λ ′ .
Still thanks to Birkhoff Ergodic Theorem, we also have

Λ ′ χdL = S * M χdL = S * M L ρ (χ)dL(ρ) = Λ ′ χdL + Λ ′c L ρ (χ)dL(ρ).
As L ρ (χ) ≥ 0 a.e., we obtain that L ρ (χ) = 0 on some Λ 1 ⊂ Λ ′c satisfying L(Λ 1 ) = L(Λ ′c ).

We pick such a subset Λ 1 in proposition 3.5. Let Λ 0 be a subset of Λ 1 verifying (1) and L(Λ 1 ) = L(Λ 0 ). Thanks to the first part of the proposition, we know that there S ⊂ N of density 1 such that any accumulation point of the sequence (µ j ) j∈S is of the form

α L |Λ ′ L(Λ ′ ) + (1 -α)ν 0 ,
where α ≥ 0 and where ν 0 belongs to Cv(Λ 0 ). We underline that the properties of Λ 0 and χ imply ν 0 (χ) = 0. Our goal is to show that there exist subsequences for which α can be chosen positive.

For that purpose, we will study the limit of the sequence (µ j (χ)) j∈S and show that it must be positive for some subsequences of positive density.

5.2.

Proof of the proposition. First, we observe that one can again replace Op by a nonnegative quantization procedure Op + . In particular, one gets, as j → +∞, µ j (χ) = ψ j , Op + (χ)ψ j + o(1), and ψ j , Op + (χ)ψ j ≥ 0 (as χ ≥ 0). Thus, without loss of generality, one can look at the accumulation points of the sequence µ + j (χ) := ψ j , Op + (χ)ψ j , j ∈ S, where S ⊂ N is of density 1. Thanks to the local Weyl law (5), we observe that

lim λ→+∞ 1 N (λ) j∈S:λ 2 j ≤λ 2 µ + j (χ) = lim λ→+∞ 1 N (λ) j:λ 2 j ≤λ 2 µ j (χ) = S * M χdL = Λ ′ χdL.
Moreover, using lemma 2.3 and the preliminary remarks, one can find a subset S ′ ⊂ S of density 1 such that any accumulation point of the sequence (µ + j (χ)) j∈S ′ belongs to the interval

[essinf L ρ (χ), esssup L ρ (χ)] = 0, Λ ′ χdL L(Λ ′ ) .
We introduce the notation

α j := µ + j (χ) Λ ′ χdL ≥ 0.
In order to prove our proposition, it remains to verify that, for every 0 < ǫ ≤ 1, there exists a subset S ǫ ⊂ S ′ of density ≥ 1-ǫ L(Λ ′ ) -1 -ǫ such that any accumulation point of the subsequence (α j ) j∈Sǫ belongs to the interval [ǫ, L(Λ ′ ) -1 ]. The proof is quite straightforward: we briefly explain it for the sake of completeness.

Fix now 0 < ǫ ≤ 1. Let η ≪ ǫ be a small positive number. From the properties of the sequence (α j ) j∈S ′ , there exists A > 0 (depending on η) such that, for λ 2 ≥ A,

1 -η ≤ 1 N (λ)
j∈S ′ :λ 2 j ≤λ 2 α j and (j ∈ S ′ and λ 2 j > A =⇒ α j ≤ L(Λ ′ ) -1 + η).

Thus, one gets, for λ 2 ≥ A,

1 -η ≤ 1 N (λ) j∈S ′ :λ 2 j ≤A α j + ǫ 1 N (λ)
♯ j ∈ S ′ : A < λ 2 j ≤ λ 2 and α j < ǫ

+(L(Λ ′ ) -1 + η) 1 N (λ)
♯ j ∈ S ′ : A < λ 2 j ≤ λ 2 and α j ≥ ǫ .

It implies that, for λ 2 large enough (depending on η and on A),

1 -η ≤ η + ǫ + (L(Λ ′ ) -1 + η -ǫ) 1 N (λ)
♯ j ∈ S ′ : λ 2 j ≤ λ 2 and α j ≥ ǫ .

In other words, we have shown that, for every η > 0, lim inf

λ→+∞ 1 N (λ) ♯ j ∈ S ′ : λ 2 j ≤ λ 2 and α j ≥ ǫ ≥ 1 -2η -ǫ L(Λ ′ ) -1 + η -ǫ ,
which implies the result.

5.3. The case of several components. In this last paragraph, we will discuss the more general setting of remark 3.6. We want to show that proposition 3.5 can be slightly improved in order to allow several subsets, i.e. |I| ≥ 2 with the notations of paragraph 3.1. Precisely, we want to verify that α can be chosen > 0 in equation ( 2) for a subsequence of positive density of eigenstates (without giving precise informations on the density of the subset). For that purpose, we use again the fact that Λ ′ contains a nonempty open ball (modulo a subset of measure 0) and we pick χ ≥ 0 a nonzero smooth function compactly supported in this open ball. As above, we can find some subset Λ 1 ⊂ Λ ′c satisfying L(Λ 1 ) = L(Λ ′c ) and L ρ (χ) = 0 on Λ 1 .

Let Λ 0 be a subset of Λ 1 verifying (1) and L(Λ 1 ) = L(Λ 0 ). As above, we consider a subset S ⊂ N of density 1 such that any accumulation point of (µ j ) j∈S is of the form given by equation [START_REF] Berry | Regular and irregular semiclassical wavefunctions[END_REF].

We can apply lemma 2.3 to this function: there exists S ′ ⊂ S of density 1 such that any accumulation point of the subsequence (µ j (χ)) j∈S ′ belongs to the interval 0, max i∈I Λi χdL L(Λ i ) .

An application of the local Weyl law (5) tells us that As µ j (χ) is "almost positive", we can apply the argument of the previous paragraph. In particular, for every ǫ > 0 small enough, we obtain the existence of a subset S ǫ ⊂ S ′ of positive density such that any accumulation point of the sequence (µ j (χ)) j∈Sǫ belongs to the interval ǫ, max i∈I Λi χdL L(Λ i ) .

As S ǫ ⊂ S, one knows that any accumulation point of (µ j ) j∈Sǫ is of the form

µ = α i∈I t i L |Λi L(Λ i ) + (1 -α)ν 0 ,
where 0 ≤ α ≤ 1, 0 ≤ t i ≤ 1, i∈I t i = 1 and ν 0 belongs to Cv(Λ 0 ). Finally, as µ(χ) ≥ ǫ and ν 0 (χ) = 0 (as L ρ (χ) = 0 on Λ 0 ⊂ Λ 1 ), we find that α must be positive.

Lemma 2 . 3 .

 23 Let a be an element in C ∞ (S * M, R). Then, there exists S ⊂ N of density 1 such that, essinf L ρ (a) ≤ lim inf j→+∞,j∈S µ j (a) ≤ lim sup j→+∞,j∈S µ j (a) ≤ esssup L ρ (a).

For more details on these assumptions, we refer the reader to[START_REF] Barreira | Pesin Smooth ergodic theory and nonuniformly hyperbolic dynamics[END_REF], sections 2.7 and 17.1.

Acknowledgements

This work has been partially supported by the Labex CEMPI (ANR-11-LABX-0007-01) and by the grant ANR-09-JCJC-0099-01 of the Agence Nationale de la Recherche. We warmly thank Stephan De Bièvre for discussions and comments related to this work, and the referee for precious advices in order to improve the presentation of this article.


Non-Photorealistic Animation and Rendering (NPAR) is a firmly established and vibrant research field within Computer Graphics, delivering new techniques for expressive rendering and visual communication of ideas and information. As a field, NPAR's origins reach back over twenty years, to seminal works exploring the emulation of traditional artistic media and the stylized visualisation and abstraction of geometry and images. Today, NPAR has diversified to cover a much broader range of topics, spanning computational photography, vector graphics, perceptual models, interactive methods to augment the creative process, motion stylization, and video enhancement. Accordingly, NPAR has become a focal point for the convergence of several disciplines; drawing not only from computer graphics, but also from computer vision, human computer interaction, psychology, and the Arts. NPAR is now a truly cross-disciplinary research field. Within this Special Section we are pleased to present extended versions of nine research papers presented at the 6 th ACM / Eurographics International Symposium on Non-Photorealistic Animation and Rendering (NPAR 2010) in Annecy, France. This prestigious symposium was established a decade ago in response to the growing interest in NPAR research, and remains a highly competitive publication forum for this specialism. The papers collected herein have been extended from their NPAR 2010 versions through addition of a further significant technical contribution and new results. All papers passed through at least one additional round of peer review, further to the doubleblind review process already conducted via NPAR 2010. Any extended paper for which one guest editor held a conflict was handled separately and independently by the other editor. Where possible, the reviewers evaluating the paper accepted to NPAR 2010 were the same invited to review the extended paper for this Special Section. We are also indebted to the additional reviewers recruited to ensure the high quality of this Special Section, and thank all of the anonymous reviewers for their thoughtful feedback.

The papers within the Special Section touch on many

Email addresses: j.collomosse@surrey.ac.uk (John Collomosse), isenberg@cs.rug.nl (Tobias Isenberg) of the topics outlined above, representing a good crosssection of current activity and trends in NPAR. One such trend is style transfer through example-based rendering, which has gathered considerable momentum through the fusion of NPAR with Computer Vision and Machine Learning techniques. "Directional Texture Transfer with Edge Enhancement" [START_REF] Lee | Directional Texture Transfer with Edge Enhancement[END_REF] describes how high-frequency textures such as brush-work, may be learned and re-applied to stylize photographs, principally in a painterly style. Visual structures are preserved and enhanced by considering both the image texture and the gradient direction in the style transfer process. "Progressive Color Transfer for Images of Arbitrary Dynamic Range" [START_REF] Pouli | Progressive Color Transfer for Images of Arbitrary Dynamic Range[END_REF] also proposes an example-based process for photograph stylization, focussing on the transfer of high-dynamic range (HDR) color rather than texture.

The best paper award at NPAR 2010 was received by the authors of "Non-photorealistic Depth-based Image Editing" who, for the Special Section, have extended their work [START_REF] Lopez-Moreno | Non-Photorealistic, Depth-Based Image Editing[END_REF] to cover additional styles and have leveraged the GPU for real-time stylization. In doing so the technique shows potential for video stylization; a perennial topic in NPAR, but especially timely in this case due to the increasing presence of NPAR in movie special effects, and the recent trend toward depth-aware post-production techniques. Progress toward temporal coherence in video stylization is presented in "Stylized Ambient Displays of Digital Media Collections" [START_REF] Wang | Stylized Ambient Displays of Digital Media Collections[END_REF]. The authors adopt a segmentation-based approach to stylization, presenting a new video segmentation algorithm. A novel NPAR application is also presented in the form of a Digital Ambient Display, situated in domestic living spaces, that stylizes and transitions through home media collections. Image stylization is also addressed in both "Contour-drive Sumie rendering of real photos" [START_REF] Ning | Contour-Driven Sumi-e Rendering of Real Photos[END_REF] and "Scale-dependent and Example-based Stippling" [START_REF] Martín | Scale-Dependent and Example-Based Stippling[END_REF]. The latter draws upon observational studies of artistic practice to guide the placement of stipple patterns, and contributes a new user study as well as a GPU implementation in this extended paper. The integration of artistic practice into NPAR research, and the evaluation of results using well-grounded studies, continues to be a principal challenge within the field; a point noted by Doug DeCarlo and Matthew Stone [START_REF] Decarlo | Visual Explanations[END_REF], Aaron Hertzmann [START_REF] Hertzmann | Non-Photorealistic Rendering and the Science of Art[END_REF], and Amy Gooch et al. [START_REF] Gooch | Viewing Progress in Non-Photorealistic Rendering Through Heinlein's Lens[END_REF] in their NPAR 2010 meta-papers outlining grand challenges in the field. It is encouraging to see a trend toward such studies in recent NPAR papers, and many of those within this Special Section.

A common practical motivation for NPAR is to improve communication and comprehension of visual content. "Multi-Perspective Compact Explosion Diagrams" [START_REF] Tatzgern | Multi-Perspective Compact Explosion Diagrams[END_REF] addresses this goal through automated generation of exploded diagrams from mesh geometry. An alternative presentation style is explored in "Shape & Tone Depiction for Implicit Surfaces" [START_REF] Brazil | Shape & Tone Depiction for Implicit Surfaces[END_REF], where algorithms for stroke placement on HRBF implicits are considered to render geometry in a Pen-and-Ink style. A rather unique approach to producing fluid effects using vector graphics is described in "Vector Graphics Depicting Marbled Flow" [START_REF] Ando | Vector Graphics Depicting Marbling Flow[END_REF], with a focus on GPU implementation.

The Special Section would not have been realised without the valued contributions of the authors, and the hard work of a number of individuals. In particular, the efforts of Prof. Joaquim Jorge both in securing this Special Section for NPAR 2010, and in his assistance in the production process. In the latter regard, our thanks are extended to Emily Rae and all those at Elsevier for an efficient production schedule. We also thank those involved

  Preface: Special Section on Non-Photorealistic Animation and Rendering (NPAR) 2010 John Collomosse a , Tobias Isenberg b,c a Centre for Vision, Speech and Signal Processing, University of Surrey, UK. b Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, The Netherlands. c DIGITEO & CNRS/INRIA, France.

  in the organisation of NPAR 2010; co-chair Morgan McGuire, the local organisers at INRIA Grenoble and Polytech Savoie, and the many individuals on the NPAR Steering Committee who facilitated this Special Section. John Collomosse, University of Surrey Tobias Isenberg, University of Groningen Co-Guest Editors of the Special Section on NPAR 2010. November, 2010.

John Collomosse is a lecturer within the Centre for Vision Speech and Signal Processing (CVSSP) at the University of Surrey. His research explores the relationships between artistic depiction and real-world imagery. Applications focus on non-photorealistic rendering of images and video, and the use of sketches to drive visual search of multimedia collections. He holds PhD in Computer Science from the University of Bath ( 2004) and is a Chartered Engineer.

Tobias Isenberg is an assistant professor for computer graphics and interactive systems at the University of Groningen, the Netherlands. Since September 2010, he also holds a Digiteo Chair in collaboration with CNRS and INRIA in Orsay, France. He works on topics in non-photorealistic and illustrative rendering and explores applications, for example, in scientific visualization. He also investigates interaction approaches for non-photorealistic and illustrative visualization, in particular, using large, touch-sensitive displays.


Wouter Lueks

Ivan Viola
email: ivan.viola@uib.no

Matthew Van Der Zwan

Henk Bekker
email: h.bekker@|isenberg@cs.rug.nl

Tobias Isenberg

Spatially Continuous Change of Abstraction in Molecular Visualization

Keywords: I, 3, m [Computer Graphics]: Miscellaneous-Scientific Illustrative Visualization, Molecular Visualization

Based on an approach for the temporal change of abstraction in molecular visualization we describe how to achieve a spatially explicit control of abstraction. This allows us to depict different abstraction stages of a single molecule in a single still-image visualization. This approach works best for long, linear molecules with repeating substructures that allow viewers to visually compare the continuous representational changes of these parts.

INTRODUCTION AND MOTIVATION

Abstraction plays an essential role in molecular visualization because the important bio-chemical processes occur at different levels of scale of the involved biological structures. Traditionally, both illustrators and software tools have employed/provided a number of well-defined abstraction types including the space-fill diagram, the balls-and-sticks model, the licorice visualization for the primary structural depiction, and backbone and the ribbon model for communicating the secondary structural arrangement within the macromolecule. While these abstraction communicate internal structure of a molecule, tertiary and quaternary structure describe the outer appearance of the molecule. All corresponding abstraction stages, however, are distinct forms of depiction that are separate from each other and whose (spatial) relations with each other are not always easy to understand. In a recent paper [START_REF] Van Der Zwan | Illustrative Molecular Visualization with Continuous Abstraction[END_REF] we demonstrated how to emphasize this spatial connection between some of the internal abstraction types by facilitating a seamless transition between the space-fill, balls-and-sticks, licorice, backbone, and ribbons abstraction stages. Moreover, we integrated this one axis of abstraction into a three-dimensional abstraction space that also allows us to explore molecular abstraction with respect to the 'illustrativeness' of the depiction and the support of spatial perception.

Nevertheless, this interactive exploration of the different forms of abstraction in molecular visualization [START_REF] Van Der Zwan | Illustrative Molecular Visualization with Continuous Abstraction[END_REF], at any given time, only provides visualizations that represent a single point in the abstraction space. Consequently, the use of interaction is essential to the approach to be able to transition over time between different stages in order to understand the relationships between them. However, in some cases it may be beneficial to be able to visualize different abstraction stages of a single molecule in a single visualization to be able to understand them without the need for animation or interaction. In this case the transition of abstraction would not happen temporally but spatially, for example along the backbone of a longer molecular structure. Molecules with repeating structures such as DNA or RNA, and certain proteins are particularly well suited for such an approach because the viewer can understand the representational changes introduced by the different levels of abstraction by comparing the depiction of the several repeating substructures of the molecule. In this way our new approach with spatially contin- Seamless spatial transition between visual abstractions is wellaligned with the principles utilized in the craft of molecular illustration. Illustrators are taking the advantage of perceptual and cognitive capabilities of the viewer, in this case the object constancy, so that multiple aspects of the molecular structure are simultaneously communicated through the resulting illustration. We adapt this illustration concept for interactive data visualization.

REALIZATION

A spatially continuous structural or illustrativeness abstraction at a single point in time requires, for each atom or bond, a parameter t s and t i such that we can determine the abstraction f (t s ) and f (t i ) for the entire molecule (following van der Zwan et al.'s [START_REF] Van Der Zwan | Illustrative Molecular Visualization with Continuous Abstraction[END_REF] notation). We determine these parameters by first identifying the molecule's backbone and then using it to define the change of structural or illustrativeness abstraction along its length. Next, we propagate these parameters to the rest of the molecule by simply assigning the specific values of t s and/or t i , at locations where a non-backbone structure connects to the backbone, to the entire non-backbone structure.

These parameters are passed to the shaders that determine the rendering of the structurally abstracted molecule as well as its illustrativeness. Even though the the parameters are only defined once per atom/bond or once per amino acid, the relatively small size of these elements when compared to the entire molecule results in viewers perceiving a gradual change of the two types of abstraction (Fig. 1). Moreover, we do not change the abstraction by means of support of spatial perception in this spatially explitit manner because of the global nature of perception-we believe that different forms of support of spatial perception in a single image would confuse viewers.

RESULTS AND DISCUSSION

Fig. 1 shows the general approach: instead of applying the same level of structural abstraction to the whole DNA molecule as in [START_REF] Van Der Zwan | Illustrative Molecular Visualization with Continuous Abstraction[END_REF], we use a higher structural abstraction level overall and apply less abstraction only to a localized part of the two strands (Fig. 1, bottom) to show that part in detail. This continuous change of abstraction can easily be understood by viewers due to the DNA's repetitive structure and further illustrates the fact that only a small segment is being shown. To illustrate the application for a less regular molecule, Fig. 2 shows an example where the spatial control of structural abstraction has been applied to different regions stracted way. Fig. 3 shows more examples of selective structural abstraction, in this case (as opposed to Fig. 1) only applied to one strand of a DNA section. However, by also using selective abstraction of the illustrativeness axis as shown in Fig. 4 we can further emphasize certain aspects of the molecule such as one strand of a DNA section. Here, simply the application of a constant (high) illustrativeness to the part of the DNA that is not in focus places emphasis on the other strand (Fig. 4, top). If we now control the illustrativeness of the strand of the DNA that is in focus similar to how we control its structural abstraction (Fig. 4, bottom) we can further concentrate the focus to the one single part shown in detail. Finally, Fig. 5 illustrates the interaction between two molecules, with the specific interaction site emphasized in low structural abstraction.

CONCLUSION

Seamless transition between visual abstractions is effective for communicating how different structural aspects relate to each other. This transition can be realized over time in an animation or spatially, e. g., along the strand(s) of the macromolecule. Further work to strengthen the communication of the hierarchical aspect of the structural arrangement in proteins, for example, can be oriented on the role of graphically embedding of one visual abstraction in its hierarchical parents.

Figure 1 :

 1 Figure 1: DNA illustrating the spatial control of abstraction.

Figure 2 :

 2 Figure 2: Proteins with structural abstraction applied in a localized fashion.

Figure 3 :

 3 Figure 3: DNA and selective structural abstraction.

Figure 4 :

 4 Figure 4: Structural abstraction combined with abstraction by means of illustrativeness.

Figure 5 :

 5 Figure 5: Interaction of a protein with another molecule.


Maarten H Everts
email: maarten@bitpuzzle.com

Henk Bekker

Jos B T M Roerdink

Tobias Isenberg
email: isenberg@cs.rug.nl

Flow Visualization using Illustrative Line Styles

We present a flexible illustrative line style model for the visualization of streamline data. Our model partitions vieworiented line strips into parallel bands whose basic visual properties can be controlled independently. We thus extend previous line stylization techniques specifically for visualization purposes by allowing the parametrization of these bands based on the local line data attributes. We demonstrate the effectiveness of our model by applying it to 3D flow field datasets.

Introduction

The flow of fluids and gases plays an important role in a wide variety of real-world phenomena. Examples include the aerodynamics of cars, the heat distribution in offices, and the airflow around falling ink droplets. Consequently, flow has been extensively studied, typically through threedimensional simulations. These simulations yield large amounts of data containing information at multiple scales; for some applications the general structure of the flow is most relevant, for others the small local deviations are the subject of study. Visual representations of flow data help in understanding its behavior and over the years a large number of methods have been developed for this purpose. Initially, most flow visualization methods employed photorealistic rendering techniques, but later-on also methods that borrow principles from scientific illustration were developed.

Inspired by such illustrative visualization techniques [START_REF] Rautek | Illustrative Visualization: New Technology or Useless Tautology[END_REF], we present a flexible method for illustratively depicting streamlines generated from 3D vector fields. We achieve this flexibility by introducing a line style model whose parameters can be interactively manipulated, thus facilitating the interactive exploration of the parameter space of visual streamline representations. This allows the user to select and generate the representations that are most suitable for the data and communication goals at hand.

In order to achieve flexible parametrization of line styles we generalize a previous illustrative approach for line visualization [START_REF] Everts | Depth-Dependent Halos: Illustrative Rendering of Dense Line Data[END_REF], by subdividing the view-oriented line strips that represent the streamlines. These strips are split into bands that are arranged orthogonally to a line's direction, and whose shape, color, relative distance to the viewer, and width can be independently controlled. In addition, we allow these line parameters to individually depend on local data attributes such as temperature or velocity.

In summary, the contributions of this paper are a flexible line style model for use in scientific streamline visualizations and a fast yet flexible implementation of this model on the GPU. We demonstrate the power of our approach for a number of 3D flow datasets that exhibit complex flow patterns.

Related Work

In this section we discuss related work in the fields of flow visualization and illustrative visualization.

Flow Visualization

Being one of the most fundamental subjects for visualization, a broad range of methods have been developed for the visualization of flow datasets. McLoughlin et al. [START_REF] Mcloughlin | Over Two Decades of Integration-Based, Geometric Flow Visualization[END_REF] survey both flow visualization in general and integration-based, geometric flow visualization in particular. A key component of integration-based flow visualization methods is the use of geometric objects to depict the properties and structure of the flow. These objects are generated by integrating over the underlying velocity field-starting from a set of seed points.

Lines are the most widely used primitives for this purpose, and in the context of steady flow such trajectories are called streamlines, whereas for unsteady flow streak and pathlines are used. The challenge for the visualization of threedimensional streamlines is overcoming perceptual problems. Simply rendering large numbers of lines can quickly lead to clutter and occlusion, not to mention the fact that the general thinness of a line makes it hard to convey depth and spatial relationships.

Solutions to deal with these perceptual challenges include careful placement of streamlines through seeding strategies (see McLoughlin et al. [START_REF] Mcloughlin | Over Two Decades of Integration-Based, Geometric Flow Visualization[END_REF] for an overview), illuminated streamlines [START_REF] Zöckler | Interactive Visualization of 3D-Vector Fields Using Illuminated Stream Lines[END_REF], and shaded tubes or ribbons [START_REF] Ueng | Efficient Streamline, Streamribbon, and Streamtube Constructions on Unstructured Grids[END_REF]. Particularly relevant for this paper are the approaches that employ textured view-oriented triangle strips to mimic shaded tubes [START_REF] Schussman | Scalable Self-Orienting Surfaces: A Compact, Texture-Enhanced Representation for Interactive Visualization of 3D Vector Fields[END_REF][START_REF] Stoll | Visualization with Stylized Line Primitives[END_REF]. Such shaded primitives need to have a certain width for the depth perception to work and have only a limited number of visual variables to convey additional information: typically only width and color, although textures can also be used to convey information about the flow [START_REF] Stoll | Visualization with Stylized Line Primitives[END_REF]. In terms of flexibility in controlling the appearance of flow streamlines, the approach by Shen et al. [START_REF] Shen | Interactive Visualization of Three-Dimensional Vector Fields With Flexible Appearance Control[END_REF] that uses 3D flow textures is relevant.

Illustrative Visualization and Line Stylization

Illustrative visualization methods [START_REF] Rautek | Illustrative Visualization: New Technology or Useless Tautology[END_REF] use and apply the principles of (scientific) illustrators to achieve the clarity and effectiveness found in traditional illustrations. Naturally, many of these methods employ techniques from the field of non-photorealistic rendering (NPR). NPR methods particularly relevant to our work aim to replicate line drawings and, for this purpose, support different line styles. Dooley and Cohen [START_REF] Dooley | Automatic Illustration of 3D Geometric Models: Lines[END_REF], for example, use dashing for illustrating geometric models, whereas the difference vectors of Schlechtweg et al. [START_REF] Schlechtweg | Surfaces to Lines: Rendering Rich Line Drawings[END_REF] permit a larger palette of styles. Other line style parametrization methods include stroke texturing [START_REF] Kalnins | Coherent Stylized Silhouettes[END_REF][START_REF] Kalnins | WYSIWYG NPR: Drawing Strokes Directly on 3D Models[END_REF][START_REF] Northrup | Artistic Silhouettes: A Hybrid Approach[END_REF], multi-resolution curves [START_REF] Finkelstein | Multiresolution Curves[END_REF], skeletal strokes [START_REF] Hsu | Drawing and Animation Using Skeletal Strokes[END_REF][START_REF] Hsu | Skeletal Strokes[END_REF], and programmable line styles [START_REF] Grabli | Programmable Rendering of Line Drawing from 3D Scenes[END_REF][START_REF] Isenberg | G-Strokes: A Concept for Simplifying Line Stylization[END_REF]. These NPR styles are typically applied to contour and feature lines of 3D objects, aim to replicate marks made by traditional tools, and-if used in an illustration-may carry a meaning (such as parts being hidden). In our work, in contrast, we use line styles to specifically visualize data properties of streamlines in a flow.

One important concept from the field of illustrative visualization important for our work is the use of halos [START_REF] Appel | The Haloed Line Effect for Hidden Line Elimination[END_REF][START_REF] Everts | Depth-Dependent Halos: Illustrative Rendering of Dense Line Data[END_REF][START_REF] Stoll | Visualization with Stylized Line Primitives[END_REF][START_REF] Tarini | Ambient Occlusion and Edge Cueing to Enhance Real Time Molecular Visualization[END_REF] to make objects (including lines) easier to discern from the background, thus improving depth perception. In the context of flow visualization, other illustrative methods related to our work include stroke-and painting-inspired visualizations of 2D flow fields [START_REF] Kirby | Visualizing Multivalued Data from 2D Incompressible Flows Using Concepts from Painting[END_REF][START_REF] Li | Illustrative Streamline Placement and Visualization[END_REF], illustrative 3D volume rendering [START_REF] Svakhine | Illustration and Photography Inspired Visualization of Flows and Volumes[END_REF], stylized streamlines [START_REF] Li | Image-based streamline generation and rendering[END_REF], as well as animated, dashed streamlines [START_REF] Laramee | Geometric Flow Visualization Techniques for CFD Simulation Data[END_REF] and dashtubes [START_REF] Fuhrmann | Real-time Techniques for 3D Flow Visualization[END_REF].

Line Styles for Visualization

As we have seen, line-based flow visualizations are problematic due to their limited number of visual variables as well as the occlusion that is introduced if more than a few line primitives are being used. We address these two major issues by introducing an extended line style model for visualization purposes.

Line Partitioning Into Line Bands

Such an extended illustrative line style model needs to increase the number of visual variables to allow the specifi-cation and parameterization of a variety of effects that can be flexibly used in visualization. For this purpose and inspired by halo-based line visualizations [START_REF] Everts | Depth-Dependent Halos: Illustrative Rendering of Dense Line Data[END_REF], we partition the line strips that are used to render the data lines into several bands (Fig. 1). By that separation we provide the granularity that is necessary to allow us to define visually distinguishable styles, each of the bands increasing the number of visual variables that can be controlled. These bands run parallel to the centerline and together define the visualization line style.

Specifically, we represent each line from the 3D dataset by a view-oriented line strip as done in many previous linebased rendering systems. This strip is subdivided into two mirrored sets of bands, one on each side of the line (Fig. 1). The three visual properties that we control per band are color, width, and distance offset w.r.t. the viewer, each of which can be controlled independently. While the distance offset is not actually a visual property, it has an effect when used as a depth-dependent halo [START_REF] Everts | Depth-Dependent Halos: Illustrative Rendering of Dense Line Data[END_REF][START_REF] Tarini | Ambient Occlusion and Edge Cueing to Enhance Real Time Molecular Visualization[END_REF]. In that case the halo line band is folded back, away from the viewer. The effect is that the visible width of the halo depends on the difference in distance between two lines w. r. t. the viewer, improving the depth perception. Therefore, our extended line style model can be seen as a generalization of the depth-dependent line halo technique [START_REF] Everts | Depth-Dependent Halos: Illustrative Rendering of Dense Line Data[END_REF].

Local Attribute Mapping

This basic line model allows us to specify a wide range of visual effects, notably by its capability to convey information about the data in the visualization by mapping data attributes such as temperature, pressure, etc. to a line's visual properties. Specifically, we control each line style band's color and width based on the value of local scalar line attributes by means of mapping functions.

For the color attribute, this mapping is encoded in conventional color maps that assign input values ∈ [0, 1] to RGB colors. We provide a selection of pre-defined color maps from which the user can choose a color map most suitable for that particular attribute type and the desired visual style. Similar to controlling the color of a band a user may adjust the width of a band to convey more information in the visualization. For example, mapping local velocity to band width yields wide bands where the velocity is high and thin bands where the velocity is low. To control this mapping, both a minimum and a maximum value can be set for the band width.

Flexible Band Shapes

The control of the line width property of a band can also be used to create bands with repeating line shape patterns such as dashes, droplets, etc. Moreover, the local density (or frequency) of these repeating shape patterns can be used to provide additional means for conveying local flow properties; this is particularly useful for the velocity property. For this purpose we employ (1) a shape mapping function and (2) a dedicated line shape attribute. A shape mapping function maps a line data attribute (∈ [0, 1]) to the width (∈ [0, 1]) of the band at that point of a line. As such, it defines the shape of a band. We combine this mapping function with a dedicated line shape attribute:

s x = x l mod 1, (1)
where l defines the size of the shape pattern on the line and x is a line attribute that is monotonically increasing along the line. The modulo operation ensures that the shape is repeated along the line. The choice of x in Equation 1 determines the local 'density' of the patterns. For example, choosing the distance along the data line to the seed point results in constant size patterns. However, choosing the integration time t makes the frequency of the patterns depend on the local velocity of the field: high velocity will result in a lower frequency of patterns (i. e., elongated patterns), providing additional means for visualizing the local velocity.

Together, the line shape attribute and the shape mapping function provide a flexible way to achieve a wide range of line shapes. Fig. 2 shows a number of examples of mapping functions and illustrates how the mapping function influences the shape of a band and, thus, the line style.

The Extended Line Style Model for Visualization

Together, the line bands with their color and width control, the means to parametrize the band width with mapping functions, and the repeating shape patterns extend the number of visual variables available for visualization of line data. Most of these visual variables can be combined in one visualization and convey multiple aspects of the streamline data in one image.

Implementation

Several design decisions of the conceptual line model were driven by implementation considerations. More specifically, because we aim for the interactive exploration of line styles even when applied to large datasets, the line style model should be suitable for implementation in shaders on modern GPUs. Our implementation consists of two parts, each implemented in a different type of shader. The first is the generation of view-oriented triangle strips (geometry shader) and the second is the application of the line style to the strip (fragment shader).

The transformation of the input lines (stored in GPU memory) into view-oriented triangle strips is done each rendering pass in a geometry shader. The width of these triangle strips is (pre-)calculated by multiplying a global scaling factor with the maximum the of line style widths. The width of a line style is calculated through a summation of the maximum widths of its bands. With the line strips in place as the 'canvas' for the line style, the next step is to apply the style model.

The actual application of the line style is done in a fragment shader. The main goal of this fragment shader is to decide which band of which line style should be applied to the fragment. To determine this it uses the position on the strip, the shape mapping functions, and the values of the relevant line attributes. Then, based on the settings for that band, the color (either from a color map or from a color pattern) and the depth offset of the fragment can be determined.

One additional aspect of our implementation is the use of templated shaders. The main reason for this is that the flexibility of our extended line style model yields a large number of options, which without templated shaders would result in a large number of expensive conditional statements in the shader. The templated shaders (implemented using the existing templating library Jinja21 allow us to flexibly include only the necessary shader code, based on the chosen style configurations. This approach has the additional benefit of making the shader used for rendering as small as possible.

Results

To illustrate the broad range of possible visual representations of lines that can be achieved with our line style model, we apply a number of different line styles to two sets of streamlines. The first set is generated from a snapshot of a numerical simulation of a heat driven cavity (Dataset 1), the other set is generated from a snapshot of a simulation of turbulent flow around a cube (Dataset 2). It is important to note that although the streamlines that we visualize here We start with the application of a single simple black-andwhite line style to Dataset 1 (see Fig. 3). This line style has two bands. The center-most band is white and fairly wide, whereas the outer band is thin and black. In addition, this outer band acts as a depth-dependent halo, although in this case it can also be considered a depth-dependent contour. The first thing to notice in Fig. 3 is how, despite the fact that no color has been used, the spatial relationships of the lines are still clear. Also, the depth manipulation ensures that collinear streamlines (e. g., the laminar flow at the bottom) blend together, emphasizing such collinear structures and yielding a crisper visualization. The close-up of the same dataset in Fig. 4 illustrates this aspect further.

The next step is to employ the visual parameters that our line style model introduces to convey additional information about the flow. Fig. 5 shows the application of a color map to streamlines, using a blue-purple color map to display velocity in Dataset 2. Again, the halo allows us to omit shading and still have good depth perception, making direct application of color maps possible without a potential shading that affects the perception of the colors.

Besides color maps, the other way our line style model can convey additional information is through the size and frequency of shape patterns. Fig. 6 illustrates how an arrow shape can be used to convey both direction and velocity in a black-and-white visualization. Shape and color maps can also be combined, as illustrated in Fig. 7, where light gray arrows are combined with a fairly wide halo to which a color map is applied. Again the size (length) of a pattern indicates the local velocity of the flow. Besides an indication of direction, the arrow shape also gives the visualization a certain feel of motion. A similar effect is achieved with the tadpole shape shown in Fig. 8 where also a color-mapped halo is used, but with a constant shape length.

Discussion

As illustrated by the results in the previous section, our parametrization of line styles allows for a wide variety of visual representations of lines, accompanied by visual variables to show additional information about the flow. In this section we discuss further aspects, observations, and limitations of our line style model. In terms of performance, we find that on a fairly modern graphics card (NVIDIA GeForce GTX 285), we can interactively manipulate the line style parameters whilst displaying fairly large datasets, facilitating the interactive exploration of different visual representations of lines. For reference, the two datasets in Section 5 consist of 2500 streamlines (2.5M vertices) and 390 streamlines (250k vertices), respectively.

An additional observation is that in our visualizations where the length of a (shape) pattern depends on the local velocity, the patterns are longer in high velocity areas. Whether this effect is intuitive seems to depend on the people who are asked and the kind of shape being used, as some people correlate high (pattern) frequency with velocity. A related observation is that when the difference in velocity is large, the shape might become too small in low velocity areas, see for example the right side of Fig. 6. Other rendering artefacts are possible, for example when (shaped) line strips overlap in a certain way, resulting in oddly shaped patterns. Also, occasionally there are small artefacts when the view-vector is parallel to the line direction, though the effect is minimal and methods exist to remedy this artifact [START_REF] Stoll | Visualization with Stylized Line Primitives[END_REF].

Finally, we presented our visualization results to a fluid mechanics expert in an informal discussion. In his initial reaction he commented on the "prettyness" of the im-ages and found the visualizations very suitable for illustration purposes (e. g., classroom usage) because they illustrate well-known phenomena very well. Interestingly though, he liked the simple black-and-white visualizations (such as Fig. 3) best, mainly because of their simplicity and expressive power.

Conclusion

We have presented a flexible illustrative line style model for the visualization of streamline datasets. By partitioning line strips into parallel bands whose basic visual properties can be independently controlled, we create a parametrization that allows us to represent a broad range of visual styles for line data visualization. This approach is combined with line attribute mapping functions for color and width to facilitate flexible line shapes and means to convey additional information about the flow.

Future work includes combining our exploration of line styles with interactive streamline seeding strategies to further improve the exploration of flow datasets for visualization and illustration.

Figure 1 :

 1 Figure 1: A view-oriented strip subdivided into a number of bands mirrored around the centerline.

Figure 2 :

 2 Figure 2: Shape mapping functions and corresponding line styles.

Figure 3 :

 3 Figure 3: A simple black-and-white line style applied to streamlines from Dataset 1.

Figure 4 :

 4 Figure 4: Close-up of streamlines with a simple black-andwhite line style applied to them. Notice how the depthdependent halo (contour) emphasizes collinear streamlines.

Figure 5 :

 5 Figure 5: Streamlines depicting flow around a cube, colored with a blue-purple color map to show velocity, combined with white halos for depth perception.

Figure 6 :

 6 Figure 6: An arrow shape mapping function applied with a simple black-and-white style. The size of the arrow indicates velocity.

Figure 7 :

 7 Figure 7: Streamlines depicted through light gray arrow shapes combined with a halos colored with a color map to depict velocity.

Figure 8 :

 8 Figure 8: Streamlines depicted with tadpole-shaped, fixed size patterns, combined with a 'halo' colored using a colormap (velocity).

http://jinja.pocoo.org/docs/

Acknowledgements

We thank Roel Verstappen and F. Xavier Trias Miquel for the datasets as well as their discussion and helpful feedback.