Samuel Thiriot
email: samuel.thiriot@lip6.fr

Jean-Daniel Kant
email: jean-daniel.kant@lip6.fr

Advances in Complex Systems

Keywords: agent-based modelling, diffusion of innovations, knowledge representation

A lot of agent-based models were built to study diffusion of innovations. In most of these models, beliefs of individuals about the innovation were not represented at all, or in an highly simplified way. In this paper, we argue that representing beliefs could help to tackle problematics identified for diffusion of innovations, like misunderstanding of information, which can lead to diffusion failure, or diffusion of linked inventions. We propose a formalization of beliefs and messages as associative networks. This representation allows to study the social representations of innovations and to validate diffusion models against real data. It could also make models usable to analyze diffusion prior to product launch. Our approach is illustrated by a simulation of iPod TM diffusion.

Why representing beliefs ?

Diffusion of innovations is an interdisciplinary field that studies "the spread of new ideas, opinions, or products troughout a society" [START_REF] Valente | Network Models of the Diffusion of Innovations[END_REF]. Rogers defines diffusion "as the process by which an innovation is communicated through certain channels over time among the members of a social system" [17, p. 11].

Several models were built to study diffusion of innovations, including multiagent based simulations, with different purposes. Explicative models aim to reach a better understanding of how individual interactions make collective dynamics appear. A great part of these models studies the decision/judgment level (adoption, opinion, perceived utility [START_REF] Ellison | Word-of-mouth communication and social learning[END_REF], payoff [START_REF] Banerjee | Word-of-mouth learning[END_REF], attitude, etc.). For instance, in the threshold model (e.g. [START_REF] Deffuant | An individual-based model of innovation diffusion mixing social value and individual benefit[END_REF]), social pressure makes individuals influenced by opinions of their neighbours. Several models also include the beliefs level, that is what individuals trust for a given object (one use "belief" rather than "knowledge" because these beliefs can be false or subjective). It is the case of models focused on informational cascades (see [START_REF] Roberts | New-Product Diffusion Models, chapter Disaggregatelevel Diffusion Models[END_REF] for a review) or in the consumat approach [START_REF] Janssen | Simulating market dynamics: Interactions between consumer psychology and social networks[END_REF]. In these models, beliefs are represented as single values or as a vector of values, and rarely aim to be matched against data collected on the field. Predictive models aim to produce an estimation of the future diffusion rate of an innovation. The well-known model, and the most used in industry, is the Bass aggregative model [START_REF] Bass | A new product growth for model consumer durables[END_REF]. It includes parameters for adoption due to media messages, adoption due to interpersonal communication and an index of market potential for the new product. It permits to reproduct the classical S-curve of cumulated adoption.

Despite of the large amount of litterature about diffusion of innovation, there still remain several problems that are not studied. The first lack resides in explicative power. Rogers [START_REF] Rogers | Diffusion of Innovations[END_REF] underlines that models are not able to explain innovation failures (sometimes due to misunderstanding of what innovations are or to incompatibility with beliefs or values). Rogers also remarks that most of the said "innovations" launched in markets are in fact incremental products. In this case people already understand what the innovation is, how it works, so the diffusion becomes quicker. Such processes cannot be modelled without representing beliefs of the population about innovations. The second lack is about predictive power. The Bass model can predict the future adoption rate of an innovation only after its launch, based on the adoption data from innovators and early adopters. But at this time, costs are already engaged (for building the product, for communication, etc.). Obvisouly, the predictive interest of the model is highly lowered. So, firms use less formal methods to test new concepts, like interviews or focus groups, which provide some insights on subjective perception and expectations about the innovation. Here again, it seems that modellers cannot avoid to represent beliefs.

Our main concern is to be able to tackle real-world cases. In this paper we study how a modeller can represent individual beliefs in an agent-based simulation. For such a simulation, we need a model for knowledge representation that is complex enough to be explicative and representative, but also simple enough to make its parameters' settings and data collection possible. We illustrate this approach with the simulation of iPod TM diffusion using beliefs collected across forums.

Model

Beliefs as associative links

Individual Associative Network

The concept of associative network has been widely used in social sciences and artificial intelligence to model beliefs: bayesian networks, causal networks, social repre- sentations represented as proximity networks, etc. A marketing methodology called the Means-end Chains Theory (MCT) [START_REF] Reynolds | Laddering theory, method, analysis, and interpretation[END_REF] proposes to formalize the perception of products as cognitive chains linking concrete attributes to perceived consequences for the individual and satisfaction of his values. As shown by the MCT, associative networks are relevant to represent the beliefs about products (an example is provided in Figure 1). These chains can be retrieved by semi-directed interviews, surveys or stastistical data analysis. Messages like advertisement or consumer reviews can also be represented as chains [START_REF] Reynolds | A means-end analysis of brand persuasion through advertising[END_REF], as shown in Figure 3.

Associative networks permit to represent several kinds of knowledge. We categorize knowledge as private, concrete or subjective a . The subjective part of information is about the innovation itself, like product attributes (links 2-7) and perceived functional consequences of the product (e.g. [START_REF] Newman | Renormalization group analysis of the small-world network model[END_REF][START_REF] Reppel | The ipod phenomenon: identifying a market leader's secrets through qualitative marketing research[END_REF]. This kind of information is received or retrieved by individuals through mass medias or interpersonnal communication. The private part of beliefs are about individuals themselves. These beliefs are more stable for an individual accross time [START_REF] Page | Assessing the predictive value of means-end-chain theory: an application to meat product choice by australian middle-aged women[END_REF]. For instance, the belief "speed → time saving" is used for all technological innovations. Private beliefs can be heuristics, like "high price → high quality". Private beliefs are provided as initial data by the modeller based on the population segmentation. The last kind of beliefs is about abstract judgments and is built by the individual itself based on its local information, as "product adopted by others". This knowledge is represented in agents by simple computational rules held by each agent.

From the modeller viewpoint, concepts in the model are a finite set C, which is created based on data collection or expert hypothesis. Sometimes two or more concepts are incompatible: an agent cannot trust both of them in the same time for the same social object. As in theory of evidence, we define frames of exclusivity called Θ X , with X ⊂ C. Some exemples of frames are: (solid, breakable), (good connectivity, bad connectivity).

Formally, we define knowledge as directed associations between concepts. Mathematically, a belief is a binary relation in C 2 . C 1 b a,t σ C 2 is the conviction held by an agent a ∈ A at time t that two concepts (C 1 and C 2 ) ∈ C 2 are associated with a given support σ ∈ Σ. The support represents the confidence of the agent on this belief (more details on support are provided below). In this model, existence of a link represents belief. No link means ignorance. Disbelief is modelled as the belief in the opposite concept. Each individual possesses his own set of beliefs; we name this set an Individual Assocative Network (IAN). Some concepts are considered as object of interest by the agents A (agents will speak about them, they want to understand them, they can take decisions about these concepts). We use a psychosocial term [START_REF] Moscovici | Psychologie Sociale[END_REF] to design these objects of common a This taxinomy follows the one provided by Audenaert and Steenkamp's studies on means-end chains theory [START_REF] Audenaert | Agricultural marketing and consumer behavior in a changing world, chapter Means-end chain theory and laddering in agricultural marketing research[END_REF], and the discussion in the field of consumer value [START_REF] Holbrook | Consumer Value : A framework for analysis and research, chapter Introduction to consumer value[END_REF], which concludes that perceived value depends both on the intrisic product properties and on the subjective perception of consumers

( σ 1 ) ( σ 1 ) (σ 1 ) (σ1 ) ( σ 3 ) (σ3) ( σ 3 ) (σ3) ( σ 2 ) (σ1 ) (σ1 )

Beliefs Revision

Insights about persuasive communication are provided by social psychology [START_REF] Moscovici | Psychologie Sociale[END_REF]. Persuasivness of a communication depends on properties of the source like credibility, expertise, self-interest, structure of argumentation, messages order, etc. No formal model exists to compute the total persuasiveness of a communication based on these parameters. However, several formalisms are available to represent beliefs and their strength, mainly with probabilities or belief functions (see [START_REF] Smets | Handbook of Defeasible Reasoning and Uncertainty Management Systems, volume 1, chapter Probability, Possibility, Belief: Which and Where ?[END_REF] for a comparative review). But, all of these models are normative and lead to results incompatible with observable evidence. They would require us to include quantitative valuation of beliefs (as probabilities or belief masses), which would make the model harder to validate, less representative and harder to manipulate. So, we developped a solution based only on the qualitative properties of beliefs.

The sources of informations are perceived as more or less credible by individuals. Broadly speaking, personal experience is stronger than other advices, themselves stronger than advertisement. We define a set Σ that contains several levels of support (in other words: credibility, certainty, revisability, strength). Each source of information is categorized by the agents in one of these levels. Levels are defined operationally to fit observations from the population and the needs of the model. Currently we work with the following levels: no credibility is used for in- We assume that a stronger source erases the previous advice, because the new source is considered to be more credible. In some cases, however, it is possible for a strong belief (acquired by direct experience) to be modified by new weaklysupported information, because individuals accept to revise old beliefs, comply with social consensus, can be convinced by a good argumentation or another reason. That's why we choose to model belief revision based on probabilities of revision between support categories p(revise|σ old , σ new ). We built this function (Table 1) based on qualitative observations. A weak support has a low probability to modify a stronger support. However in long term, this probability becomes higher and higher, leading to invalidate old beliefs. This model is easier to validate than a quantitative representation of strength.

Retrieving from memory

We need to be able to retrieve the representation of a social object contained in an IAN. Retrieving a representation is an spreading activation process: start from the social object, then browse all the links connected to this node to build the representation R a,t o . We assume an activation propagation inspired by evidence networks: the activation strength of a concept for an object is the strength of the weakest link in the chain that links the social object to this concept. When activation follows a link, activation is filtered by the belief strength. For instance in Figure 1, activation of concept "time saving" for the social object "iPod" is "indirect experience", which is the lowest support in the chain (σ 2 ). If a node receives several levels of activation from its parents, the stronger activation is kept (MAX-activation, which is also an OR logical interpretation). In the example of Figure 1,"ease of use" has a support of "personal experience". As a result, the activated representation contains the beliefs activated and their support. In the particular case of incompatible beliefs, the activation process only keeps the strongest belief. For instance in Figure 1, the frame of exclusivity θ complexity = {ease of use, hard to use} forbids both of these concepts to be trusted at the same big audience small audience interactive topic choice forums, search on internet face-to-face static topic press, advertisement, direct experience weblogs Table 2. taxinomy of channels time. The spreading activation process sets a low activation to "hard to use" and an higher to "ease of use", so only the last one will be included in the activated representation. As shown before by the agent-based modelling community, the social structure has a huge impact on the system dynamics (e.g. [START_REF] Valente | Network models of the diffusion of information[END_REF]). A model of communication, that it too simple -like random meeting or cellular automata -doesn't seem adequate. As a consequence, we detail here explicitely the channels that support communication, the structure of messages themselves, and the topics (social objects) agents are talking about.

Communication

A channel is a support of communication that transmits information from an information source to an audience. Historically mass media were controlled by firms for persuasive communication, while interpersonal channels were only used for uninterested communication. Today individuals' reviews through specialized websites or forums could challenge traditional mass media, and interpersonal communication begins to be modified by individuals who are paid to propagate positive recommandations. To take this evolution into account, we propose to categorize channels based on their audience size and the determination of topics (Table 2). An unidirectional channel will always have a static topic (because the information source communicates about the object of its choice) while bidirectionnal channels allow interactive choice of topic. Modelling interactive topics implies modelling information research, and not only passive information reception.

11, 2008 17:5 WSPC/INSTRUCTION FILE croyances˙et˙innovation˙journal

Using associative networks to represent adopters' beliefs in a multi-agent model of innovation diffusion 7 "The Apple ipod (20gb) is certainly one of the best mp3 players available today. iPod supports up to 320kbps and even supports wav files for maximum quality... The only drawback to the sound quality is the sub-standard headphones which ship with product.

The iPod supports folders, excellent pc connectivity including USB 2.0 and Firewire support. The battery will last for around 8-10 hours. The product has NO support for protected WMA files." A mass channel is connected to a great number of agents. The agent exposure defines its probability to receive messages through this channel. An interpersonal channel represents the fact that two individuals can exchange information with a given exposure parameter (the probability for the agents to meet). A statictopic channel will only transmit passively messages, so the topic is determined by the information source. An interactive-topic choice channel asks both agents which topics they want to discuss about (the salient social objects set of each agent) and pick up randomly a social object in the union of the two sets.

Messages

Each transmission of information (either from mass media or interpersonal) is a message. A message is intended to transmit information b about a social object. A message is sent by a sender over a channel; audience will be determined by the channel itself. A communication campaign is composed of several messages broadcasted on channels during a given period.

The content of a message is a transmissible associative network (TAN), which is made of associative links (see Figure 3 for example). A TAN typically embodies only a representation of a single social object. Sometimes -especially in the case of co-branding -the network can include several social objects and their associated representations. A TAN transmitted by an extrinsic information source is provided by the modeller. A TAN from an intrinsic information source is dynamically built by this agent.

Agents

A consumer agent represents a unit of adoption. It embodies a belief base, a list of currently salient social objects and is linked to an agent profile. An agent profile contains the default exposure to mass channels, background knowledge, subjective b This definition of a message is voluntarily simplified to fit the frame of this paper. A message, especially an advertisement, also embodies some non-semantical components. production of knowledge. It also contains functions to evaluate attractivity and decide adoption. It can also include some rules to create the subjective knowledge based on local information. For instance, the fact that others have adopted a product (belief number 1 in Figure 1) is modelled by a threshold on the observed relashionships that possess the product. The definition of the agent's behavior is out of the scope of this paper. The modeller can implement whatever models he wants based on the internal representation of beliefs, which provides both beliefs and their strength. Several models exist to describe attitude formation or adoption based on beliefs, as the Theory of Planned Behaviour [START_REF] Ajzen | The theory of planned behavior[END_REF], the Fishbein model or any multi-criteria model. As an example we currently use the behavior process represented in Figure 4. We designed multicritera functions to compute attractivity and adoption, which take into consideration the support of beliefs.

Based on these three functions, the following process appears, which is compliant with existing models of buying steps or adoption process [START_REF] Rogers | Diffusion of Innovations[END_REF]. (1) first the agent becomes aware of the innovation, and receives prior information. (2) if the information is attractive enough, the agent decides to look for it (3) if the agent thinks it has enough information, it decides to adopt or not (4) using the product, it receives more information by usage and participates to word-of-mouth.

Application to iPod TM

Data collection

We retreived data from the published means-end chains analysis of iPod TM [START_REF] Reppel | The ipod phenomenon: identifying a market leader's secrets through qualitative marketing research[END_REF] and from statistical analysis of reviews provided by consumers on specialized websites. This data is used to determine the content of interpersonal messages and to insert background knowledge into agents. Associative networks permit to represent background knowledge. For instance in Figure 1, the links (9,16) represent fears of late majority about technology: it's hard to use and leads to waste of time.

We identified the following static-topic mass channels: TV advertisement, generalist and specialized press, experience with the product. We set exposure to each media from general statistics published about TV ads exposure, press reading, etc. We used as a social structure a small-world graph (a regular lattice with shortcuts as proposed in [START_REF] Newman | Renormalization group analysis of the small-world network model[END_REF]). The exposure level to social interactions is retreived from a study [START_REF] Carl | What's all the buzz about? everyday communication and the relational basis of word-of-mouth and buzz practices[END_REF] about word-of-mouth, which quantifies on average 15 word-of-mouth

Agent profiles

We adopt the classical segmentation used in diffusion of innovations. Innovators like what is new, fun. They enjoy to spent time to learn how an innovation works. They are able to understand technological terms. They read specialized press nearly ones a week. They are more impulsive than others, and can adopt an innovation as soon as it is available. They easily speak about innovations. They like to be alone to possess new things, and an innovation already possessed by others loose its interest. Early adopters sometimes read specialized press. They like new thinks, they carefully study available information before buying. Individuals from early majority like to be on-trend, with new products. They already have a good knowledge about technology, but like to have feedback from first adopters before buying. Late majority don't cares about the novelty of a product. They focus on the utilitarian aspect, don't like to loose time to learn new techologies. As part of their background knowledge, they believe that technological innovations are hard to use (as represented by beliefs 9,16 in Figure 1). They consider a piece of information as true only if it comes from someone else with direct experience. Laggards have a low exposure to press, and retreive most of their information from interpersonal communication.

Simulation

The model is implemented with the Repast Framework. In this discrete-time simulation, each step represents one week.

Graph 5 shows the output of the model. Awareness starts before adoption due to annoucement information transmitted about iPod TM . Because an announcement is only transmitted in specialized press, mainly innovators and early adopters are aware of the product and can propage word-of-mouth around them. Then the product is launched, with information in generalist press and TV advertisement. All the population becomes aware of the product and can adopt it. Early majority requires indirect information from previous users or independant reviews to adopt. Late majority needs indirect feedback to adopt. The last curve in this figure shows that the diffusion is made quicker if another media (here: internet) permits to retrieve others advices quicker than face-to-face communication; this media is highly efficient because it permits to determine interactively topics and to retrieve credible information.

Observations

How to improve diffusion ? In this model advertisement on its own doesn't lead to adoption, but can make the product salient in individual minds and provoque adoption or word-of-mouth. The best idea to make diffusion quicker is to facilitate word-of-mouth, which is required to persuade late majority and laggards to adopt. A good timing, and attractive information, is require to stimulate word-of-mouth. If new information is sent when individuals are still looking for information, then this new information will be transmitted quickly through interpersonnal communication. Observability, one of the factors mentionned by Rogers, also facilitates diffusion in this model. In the case of iPod TM , the white hearspeakers are easily identifiable, and are related to iPod TM based on advertisement campaigns. So potential adopters are aware of others adoption, leading them to follow this indirect recommandation. The importance of usage value, as in reality, is confirmed, because individuals who use the products are highly credible and can provoque adoption; it is of prime importance that they be satisfied by the product.

Social representations of the innovation appear in the model. In the beginning of diffusion, we can observe collective representations shared by several agents (Figure 6): individuals who have already adopted possess a large amount of information provided by experience, while others only have a representation created from advertisement. Individuals who had no knowledge about mp3 players discover through word-of-mouth what are the criteria for evaluating the innovation. While late majority is initialized with no knowledge about mp3 players, all individuals end with general considerations about autonomy or storage capacity. We also observe examples of incomprehension: an individual who has no knowledge about storage capacity is unable to understand what "10Gb" means, but he will learn it through interper- iP od is held by several agents who just have information from advertisement, and know that iPod TM is already widely adopted.

sonal messages or well-designed advertisement (with the slogan "1000 songs in your pocket").

Discussion

Representing knowledge as associative networks permits to create models which can be tested against real data, and to represent both messages and individual knowledge in a computationnally tractable way. This representation is highly representative and manipulable, even for non-experts. Implicitly it allows to model misunderstanding of information, word-of-mouth or launch of related innovation in a more plausible way -in fact models that were expected by Rogers. Hence we could build models that represent the whole adoption process, from awareness to decision.

When a diffusion model is built to be used as a decision-support system, this approach is obviously more instructive. Through simulation the modeller is able to study the true parameters of diffusion of innovation (those mentionned by Rogers and used by marketers): what is the perception of products ? In what way are consumers aware of a product ? What is the background knowledge of individuals, and will they be able to understand information ? Why does an innovation provoke word-of-mouth ? Used before innovation launch, the model can be parameterized from interviews -for subjective perception of the innovation -and general information-for background knowledge -, giving one an efficient methodology to test possible diffusion cases.

The main limitation of such a model is the state of knowledge about human behavior and social phenomena: no sufficient information is available about the structure of real social networks, what provokes word-of-mouth, etc. Our future work will be focused on the validation of models based on associative networks, including interview protocols and statistical methods.

Fig. 1 .

 1 Fig.1. Exemple of Individual Associative Network (IAN) retrieved by interview for iPod TM (study on what people like or dislike for this product). To improve lisibility only useful supports are provided. σ 1 represents the support "personal experience", σ 2 means "indirect experience" and σ 3 "no credibility"

Fig. 2 .

 2 Fig. 2. Communication in a real population

Fig. 3 .

 3 Fig. 3. Transcription of a consumer advice retreived on a website (left) as a TAN (right)

Fig. 4 .

 4 Fig. 4. Exemple of decision process

  Using associative networks to represent adopters' beliefs in a multi-agent model of innovation diffusion 9 episodes per week.

Fig. 5 .

 5 Fig. 5. Simulation of iPod TM diffusion in a population of 5000 agents

Fig. 6 .

 6 Fig. 6. Example of social representations. The left representation SR earlyadopters,12 iP od is shared by first adopters who have already an experience with iPod TM . The right representation SR latemajority,12

Table 1 .

 1 Using associative networks to represent adopters' beliefs in a multi-agent model of innovation diffusion 5 formation from advertisement, plausible is used for advice from someone, indirect experience represents feedback of someone based on its personal experience. Personal experience represents the strongest level for beliefs acquired by the agent direct experience. probability of revising a belief based on the support level of the previous belief σ old(top) and on the support level of the new information σnew (left column)

	January	11,	2008 17:5 WSPC/INSTRUCTION	FILE
	croyances˙et˙innovation˙journal
				no credibility	plausible	indirect experience	personal experience
		no credibility		0.9	0	0	0
		plausible		1	0.9	0.01	0.001
		indirect experience	1	1	0.9	0.001
		personal experience	1	1	1	0.9


Marek Dvorožňák

Czech

Oliver Wang

Example-Based Expressive Animation of 2D Rigid Bodies

Despite the recent success of computer-generated animations, traditional hand-drawn approaches often yield more expressive and stylized looks than those produced with the currently available digital tools. However, creating hand-drawn animations is a tedious

Optionally, the final appearance of the drawings can be modified by re-synthesizing di erent stroke textures (d).

We present a novel approach to facilitate the creation of stylized 2D rigid body animations. Our approach can handle multiple rigid objects following complex physically-simulated trajectories with collisions, while retaining a unique artistic style directly speci ed by the user. Starting with an existing target animation (e.g., produced by a physical simulation engine) an artist interactively draws over a sparse set of frames, and the desired appearance and motion stylization is automatically propagated to the rest of the sequence. The stylization process may also be performed in an o -line batch process from a small set of drawn sequences. To achieve these goals, we combine parametric deformation synthesis that generalizes and reuses hand-drawn exemplars, with non-parametric techniques that enhance the hand-drawn appearance of the synthesized sequence. We demonstrate the potential of our method on various complex rigid body animations which are created with an expressive hand-drawn look using notably less manual interventions as compared to traditional techniques.

CCS Concepts: • Computing methodologies → Motion processing; Nonphotorealistic rendering;

process that requires years of training by an artist and countless hours of labor. Furthermore, style is a highly personalized concept, and two di erent artists never animate exactly in the same way. As a result, example-based stylization has been a long-standing goal in computer graphics.

In this work we focus on rigid bodies, which are particularly challenging to animate by hand, since multiple objects may collide and rebound in ways that are di cult to plan in advance. Conversely, using physical simulation, computer-based methods can quickly give rise to rigid body animations with realistic trajectories, but ones that lack expressiveness. Our main goal is therefore to combine the ease of use of computer-simulated 2D rigid body animations with the expressive qualities of hand-drawn techniques.

To accomplish this goal, we have a number of added requirements. First, editability is of paramount importance to animators, and an ideal solution should work iteratively, always providing the artist with the ability to re ne the current solution. Second, producing each hand-drawn frame is time consuming, so a practical examplebased 2D animation system should be able to generalize from a very limited set of artistic inputs, while being able to apply these edits seamlessly into the dense set of nal target frames. These two requirements make example-based 2D animation out of reach of current data-driven machine learning techniques, due to the scarcity of data (tens of exemplars, rather than tens of thousands), and uniqueness of each style.

Instead, our approach is inspired by a work ow that is widespread among both traditional and digital animators. A 2D animation is successively produced and re ned at three di erent temporal scales (see Figure 3): the full animation scale at which timing, contacts and trajectories are planed; the pose-to-pose scale, at which the overall dynamics and deformations between contacts are considered; and the frame-to-frame scale at which the actual drawings with secondary deformations and precise collisions are produced.

At the full animation scale, we split the input computer-generated sequences based on collision events, and we independently analyze and stylize each sub-sequence around a hit point, which we call a key pose. Next, for every sub-sequence, we estimate the spatial deformations of the hand-drawn exemplars before and after the key pose; these are then transferred using a parametric synthesis algorithm. Correspondences between sub-sequences are estimated by leveraging the physical properties of each frame, which ensures preservation of appropriate stylistic e ects given the forces applied at each frame. The nal frame-by-frame drawings are then synthesized from the artist drawings using a non-parametric technique, that captures residual deformations and appearance details.

We show that this organization is necessary to capture both the long-and short-range stylistic choices made by the artist, while producing results that have the desired hand-drawn look and feel. When taken as a whole, our method considerably reduces the amount of work needed to create the entire sequence by hand.

In summary, we present the following contributions:

• a careful analysis of traditional hand-drawn animations, especially focusing on the correlation between physical parameters and deformations, • a parametric motion synthesis algorithm capable of transferring deformations from exemplars, • an example-based non-parametric stylization technique capturing the ne-scale drawing appearance.

PREVIOUS WORK

We present prior work related to general stylized computer animation, followed by those driven by physical simulations, and nally example-based solutions.

Techniques inspired by the principles of animation. From its beginning, one of the goals of computer graphics has been to reproduce the expressiveness of traditional hand-drawn 2D animations, while reducing the cost and e ort of producing it. The fundamental principles of animation, developed from the late 1920's to the 1930's at Walt Disney Studio [START_REF] Thomas | The illusion of life : Disney animation[END_REF], play a crucial role in this expressiveness, and many works have tried to adapt them to digital animation tools. [START_REF] Lasseter | Principles of Traditional Animation Applied to 3D Computer Animation[END_REF] describes these 2D principles -including squash and stretch, timing and spacing, anticipation and follow-through, arc trajectories and lines of action -and how they can be manually applied by an artist to produce expressive 3D keyframe animations.

Subsequent work has aimed at fully or partially automatizing those e ects. [START_REF] Wang | The Cartoon Animation Filter[END_REF] describe a simple temporal lter that produces anticipation and follow-through as well as squashand-stretch deformations by delaying parts of an existing animation, represented by 2D polygonal shapes or motion captured (MoCap) data. [START_REF] Lee | CartoonModes: Cartoon stylization of video objects through modal analysis[END_REF] obtain similar e ects on segmented objects in a video by relating a set of representative 2D deformations to the modal analysis of the object motion. Focusing on 3D skeletal animation, [START_REF] Noble | Automatic Expressive Deformations for Stylizing Motion[END_REF] present a tool to automatically bend the limbs of a character following lines of actions or arcs. On the artistically controllable side, [START_REF] Li | Stylizing Motion with Drawings[END_REF] present a sketching interface that allows a user to guide the deformation of both the input animated skeleton and surface mesh to improve the expressiveness of MoCap data. Recently, several 2D sketching systems [Kazi et al. 2014a[Kazi et al. ,b, 2016;;[START_REF] Xing | Energy-Brushes: Interactive Tools for Illustrating Stylized Elemental Dynamics[END_REF] have been developed to simplify the production of dynamic illustrations, reproducing most principles of character or special e ects animation. However these principles are essentially encoded as scripted deformations or animated loops triggered by events, and unlike our work are not tailored to the speci c style of a given artist.

Physics-driven approaches. Physical simulation is a convenient way to automatically animate a large number of 2D or 3D bodies, but expressiveness of the resulting motion is restricted by the degree of complexity modeled by the physical system. For instance, it is common to restrict the simulation to rigid bodies for computational e ciency, while traditional hand-drawn animated objects more often resemble deformable bodies governed by exaggerated physical laws. To enhance basic 3D simulations, multiple works [START_REF] Chenney | Simulating Cartoon Style Animation[END_REF][START_REF] Garcia | A Physically Based Deformation Model for Interactive Cartoon Animation[END_REF][START_REF] Haller | Non-photorealistic Rendering Techniques for Motion in Computer Games[END_REF] derive automatic procedural rules to generate squash-and-stretch and temporal e ects based on motion parameters (velocity, acceleration, etc.) but such methods have limited art-directability.

To allow artistic control, the spacetime [START_REF] Witkin | Spacetime Constraints[END_REF]] and dynamic [START_REF] Barzel | A Modeling System Based on Dynamic Constraints[END_REF] constraint formulations cast physical simulation as a constrained optimization problem. Through those constraints, the artist can direct the simulation to act as a physically-plausible interpolation mechanism between key-poses. [START_REF] Bai | Artist-directed dynamics for 2D animation[END_REF] leverage this idea to build a 2D animation system that combines keyframing of local deformations with physical simulation for powerful inbetweening. Although this approach allows an unprecedented level of artistic control and manages to reproduce many of the principles of animation, it requires the user to specify control handles, which are constrained and unnatural to use when compared to simply drawing frames, in particular when the artist desires a precise control over shape outlines.

Example-based methods. This family of techniques provides a natural and intuitive interface, where examples are used to capture the style and intent of an artist. Such approaches have already produced impressive results for static images and videos, either using non-parametric texture synthesis [START_REF] Bénard | Stylizing animation by example[END_REF][START_REF] Fišer | StyLit: Illumination-guided Example-based Stylization of 3D Renderings[END_REF][START_REF] Hertzmann | Image analogies[END_REF][START_REF] Lu | HelpingHand: example-based stroke stylization[END_REF]] or more recently with neural networks [START_REF] Gatys | Image style transfer using convolutional neural networks[END_REF]]. Yet these methods are mostly restricted to appearance stylization, leaving motion largely untouched.

There are some exceptions, such as [START_REF] Bregler | Turning to the Masters: Motion Capturing Cartoons[END_REF] who propose to capture and re-target motion from existing cartoon animations by combining a global a ne deformation with drawings interpolation using a key-shape model. [START_REF] Jones | Dynamic sprites: Artistic authoring of interactive animations[END_REF] follows a similar approach, connecting the navigation in a simplicial complex [START_REF] Ngo | Accessible Animation and Customizable Graphics via Simplicial Con guration Modeling[END_REF]] with events of a 2D physical simulation. Posespace interpolation can produce impressive results, but the quality of the output is highly dependent on a good choice of the key-shapes which an artist has to select and organize manually beforehand. To guide or enrich 3D simulations, example-based approaches augment the simulated objects with examples of desirable deformations [START_REF] Coros | Deformable Objects Alive![END_REF][START_REF] Jones | Example-based Plastic Deformation of Rigid Bodies[END_REF][START_REF] Koyama | Real-time Example-based Elastic Deformation[END_REF][START_REF] Martin | Example-based Elastic Materials[END_REF]]. In those approaches, however, exact correspondences between deformation exemplars are known beforehand and only a simple parametric deformation with limited number of degrees of freedom is used. Even though this setting may be natural for digital 3D artists, it is again limited and constraining for traditional 2D animators.

Closest to the traditional animation pipeline, Xing et al.

[2015] present an interactive system for computer-assisted 2D animation. It combines discrete texture synthesis with an as-rigid-as-possible deformation model to predict and interpolate drawings based on the current and previous frames. This approach is convincing for frame-by-frame animation, but the spatio-temporal locality of the analysis makes it unsuited for pose-to-pose planning. Since the interpolations are solely based on the past drawings using local a ne transformations, the predicted motion and deformations tend to be unnatural and cannot easily be edited, unless the artist draws most intermediate frames.

OVERVIEW

Similar in spirit to Image Analogies [START_REF] Hertzmann | Image analogies[END_REF], our algorithm transforms a target 2D rigid body animation F T into an output stylized version F O using an example-based transformation de ned by a set of source sequences F S and a corresponding set of hand-drawn exemplars F E (Fig. 2). Sequences F S and F T can be computer-generated using, e.g., physical simulation. The style exemplars F E are created by an artist digitally or physically, by redrawing a small subset of the source frames F S . In one application, the untouched frames of F S can be added to F T , in which case our method can be seen as an interactive style propagation tool. This is shown in the accompanying video where the artist rst draws over a few frames, sees the intermediate result, identi es parts which have not been successfully stylized, provides additional examples, and iterates this procedure until she is satis ed by the stylized result.

The key challenge here comes from the fact that F T will typically not contain sub-sequences exactly like those in F S , and thus stylized frames from F E cannot simply replace original rigid frames in F T . To tackle this problem, we take inspiration from guidelines in traditional 2D animation books [START_REF] Thomas | The illusion of life : Disney animation[END_REF][START_REF] Williams | The animator's survival kit[END_REF], especially from Richard Williams' hierarchical decomposition of the animation process (see Figure 3). We identify three main stages or temporal scales: (1) the full animation scale, at which timing and spacing are planned by choosing the key events, (2) the pose-to-pose stage, at which the main dynamics and deformations are de ned between two key poses by drawing "extremes" and "breakdowns", and (3) the frame-to-frame scale, corresponding to nal straight-ahead "runs" (or drawing passes) during which subtle variations and details are added.

Each of the three stages need to be analyzed for transferring the style of a hand-drawn animation and our method thus follows this organization. First, timing and spacing are speci ed by the input sequences F S and all animations are subdivided into overlapping sub-sequences around key events (Section 4). The style pairs F S : F E are then decomposed into a coarse geometric deformation D and a ne-scale "residual" stylization R (Section 5.1). Our aim is to transfer both D and R to the target sequence F T . For each target sub-sequence independently, a new parametric deformation is synthesized by selecting and blending together multiple deformations D coming from similar sub-sequences of the style pairs (Section 5.2). Finally, sub-sequences are blended together, the ne-scale details are reintroduced on a frame-by-frame basis by morphing the residual changes R, and the appearance of individual strokes is changed to have a desired look of particular artistic media (Section 6). In the following, we use the classical "bouncing ball" animation to illustrate the various steps of our algorithm; results on more complex sequences (collisions between objects, bouncing square, textured objects) are shown in Section 7 and the supplemental material.

F F 1 F 2 M 1,2 e 1 e 2 N 1
Fig. 4. Decomposition into sub-sequences -an input sequence F is subdivided into sub-sequences (F i ) of N i frames around key events (e i ∈ E) with an overlap of M i,i +1 frames with the following sub-sequence.

TIMING AND SPACING

The source F S and target F T input sequences consist in 2D rigid body animations produced using physical simulation. In practice, we use Box2D [START_REF] Catto | Box2d -a 2D physics engine for games[END_REF]]. Similar to prior work [Kazi et al. 2016], an arbitrary target rigid body is represented by its proxy geometry, e.g., bounding circle, square, or any other closed polygon. In addition to images, the simulation generates, at each frame and for each object, a set of physical parameters including the object velocity and the rotation angle α around the object centroid with respect to the local trajectory frame. The timing and spacing are dictated by the simulation; the artist draws over existing frames, establishing one-to-one temporal correspondences between F S and F E . The simulation also identi es frames at which semantically important events E occur, such as contact points or collisions. Following the guidelines of Richard [START_REF] Williams | The animator's survival kit[END_REF], these frames represent key poses. Analyzing the hand-drawn exemplars F E , we also observed that those frames, and their immediate neighbors in time, are the ones most stylized by the artist, whereas distant frames are less modi ed. In addition, we noticed that the physical parameters along the simulated trajectories before and after these key events largely in uence the artist's stylization choices, e.g., the magnitude of the deformation.

These observations motivate us to subdivide F S , F T and F E into a set of smaller overlapping sub-sequences F S i , F T i and F E i around every key event e i of E. Each sub-sequence F i contains N i consecutive animation frames and overlap with the next sub-sequence on M i,i+1 frames. As shown in Figure 4, the overlapping part between two events resides at frames where there are no abrupt changes in physical parameters and moderate artistic stylization, making them most suitable for stitching.

POSE-CENTERED DEFORMATIONS

At this stage, we consider each sub-sequence independently, and focus on the coarse deformations used by artists when hand-animating rigid bodies to reproduce e ects described in the principles of animation (squash-and-stretch, arc trajectories, lines of action). Residual deformations and appearance variation that are not captured by this coarse deformation will be reintroduced in Section 6.

Parametric deformation analysis

For each frame of a style pair F S i : F E i , we rst estimate a coarse parametric deformation D (see Figure 5(a)) using the registration algorithm of [START_REF] Sýkora | As-Rigid-As-Possible Image Registration for Hand-Drawn Cartoon Animations[END_REF] which aligns bitmap images with an as-rigid-as-possible grid deformation. However, instead of the deformable grid matching, we use a single quadratic transformation as in [START_REF] Müller | Meshless Deformations Based on Shape Matching[END_REF]. The main advantage of this quadratic model is that besides shear and stretch, it also captures twist and bending modes (see Figure 6) which better represent the larger scope of deformations used in traditional animation.

= D D -1 f s f e f r (
The output of the image registration phase consists of 12 parameters describing the corresponding quadratic deformation that warps pixels p = (x, ) from the source frame f s ∈ F S i to match pixels p = (x , ) of the stylized frame f e ∈ F E i :

x = a 11 x + a 12 + q 11 x 2 + q 12 2 + m 1 x + t 1 = a 21 x + a 22 + q 21 x 2 + q 22 2 + m 2 x + t 2 (1)
Written in matrix form:

p = A Q m t D p (2)
where p = (x, , x 2 , 2 , x , 1) is p expressed in extended homogeneous coordinates, and D is a quadratic transformation matrix composed of a ne A, purely quadratic Q, mixed m, and translation t parts:

A = a 11 a 12 a 21 a 22 Q = q 11 q 12 q 21 q 22 m = m 1 m 2 t = t 1 t 2 .

Parametric deformation synthesis

Based on traditional hand-drawn animation resources as well as our own observations and discussions with 2D animators, we make the key hypothesis that deformation is closely tied to motion. As a result, to perform the deformation transfer, we search for correspondences between source F S i and target F T j sub-sequences using physical parameters that describe the frame's motion (velocity, trajectory orientation, and the object's rotation), and we assume that the matching sub-sequences should undergo similar deformations D as the source ones. Source-target sub-sequence matching. Practically, we de ne the following di erence metric between a source sub-sequence F S i and a target sub-sequence F T j :

Di (F S i , F T j ) = λ vel Vel(F S i , F T j ) + λ dir Dir(F S i , F T j ) (3) + λ rot Rot(F S i , F T j ),
where weights λ vel , λ dir , λ rot are used to balance the in uence of individual terms:

• Vel(F S i , F T j ) measures the di erence between rigid body centroid velocities :

Vel(F S i , F T j ) = N n=1 || n (F S i ) -n (F T j )|| 2 , (4)
• Dir(F S i , F T j ) penalizes discrepancy of the trajectory orientation δ :

Dir(F S i , F T j ) = N n=1 ||δ n (F S i ) δ n (F T j )|| 2 , (5)
where computes the smallest di erence between two angles, • Rot(F S i , F T j ) accounts for di erences in the rotation α of the rigid body around its centroid:

Rot(F S i , F T j ) = N n=1 ||α n (F S i ) α n (F T j )|| 2 . (6)
When computing the metric we assume that both sub-sequences are centered at a key event and have the same number of frames N . This can be done by resampling the original trajectories to have equidistant samples according to their arc length. The longest subsequence is trimmed to have the same length as the shortest one.

Deformation blending. Since it is unlikely that any source subsequence perfectly matches a given target sub-sequence F T j , we retrieve K nearest neighbor sub-sequences F S 1 . . . F S K instead of a single one. For each frame in F T j , we then compute its stylized version as a combination of K quadratic transformations D 1 . . . D K from the K best corresponding frames in source sub-sequences using weights w 1 . . . w K proportional to their similarity:

w k = 1/Di (F S k , F T j ) K κ=1 1/Di (F S κ , F T j ) , k ∈ [1 . . . K]
where normalization is used to obtain the partition of unity. See Figure 7 for an overview of this blending scheme (with K = 3) which adds robustness to the matching process and gives more stable results than simply using the single best match (K = 1).

To perform a meaningful interpolation of the rotational part of the transformation, the a ne part A of the matrix D is factorized using polar decomposition [START_REF] Higham | Fast polar decomposition of an arbitrary matrix[END_REF]] into a linear stretch U (not used directly) and a rotation matrix R α , from which the rotation angle α = arctan(r 11 /r 21 ) is extracted. A weighted blend is computed on α 1 . . . α K : where ⊕ computes a weighted average of circular quantities. The remaining coe cients of rotation-free quadratic transformations

α = w 1 α 1 ⊕ . . . ⊕ w K α K (7) D 1 D 2 D 3 w 1 w 2 w 3 D f e 1 f s 1 f e 2 f s
D 1 • • • D K are similarly computed: D = w 1 D 1 + . . . + w K D K (8)
where D = R -α D. Finally, the blended quadratic transformation matrix D is constructed from α and D :

D = R α D (9)
Data augmentation. To generate plausible results even when the global orientation or scale of the target trajectory departs considerably from the available exemplars, we enrich the set of source analogies by scaling, rotating, and ipping the input sequences. We can directly extract the set of required rotation angles γ by analyzing the target simulation. Based on our experiments, we also use 5 scaling factors ρ between 0.2 to 1 and allow symmetries with respect to the vertical axis only (to preserve gravity e ects). For rotationally symmetric objects of order n, we modify the operator in a way that it outputs zero di erence for angles k 360 • n where the apperance of the rotated object is the same. For the circle the order of rotational symmetry is in nite so instead we set λ rot = 0.

The drawback of the augmentation is that it may lead to incorrect stylizations when the source exemplars are far from the target motion. For example, a source exemplar corresponding to a small jump will not be equivalent to a source exemplar with a higher jump. To account for this, we dampen the resulting quadratic transformation D by computing a weighted blend of D with the identity matrix I using weight ξ , proportional to the ratio of the average source and target velocities:

D = ξ D + (1 -ξ ) I with ξ = N n=1 n (F S i ) N n=1 n (F T j ) (10)
However, if rotational invariance is not (even approximately) satised, orientation augmentation cannot be used, and the artist will need to prepare a set of additional exemplars corresponding to the correct rotational motion.

, f r 2 , f r 3 producing their deformed counterparts f r 1 , f r 2 , f r 3 .
Those are then blended together using n-way morphing [START_REF] Lee | Polymorph: Morphing Among Multiple Images[END_REF]] to produce the resulting frame f r .

FRAME-BY-FRAME STYLIZATION

Although the parametric transformations D capture most of the dominant global deformations, there are still small residual deformations and appearance variations R (e.g., sketch lines of the drawings) which cannot be simply described by the quadratic deformation model. These residual changes represent a very important part of the stylization, as they provide much of the uniqueness of traditional hand-drawn, as opposed to computer-generated animation.

Extraction of the residual. Due to the parametric deformation model, extracting R from the source exemplars is straightforward. We compute and store the residual frames in F R i by "rectifying" the example frames in F E i using the inverse transformation to the deformation D estimated in Section 5.1 (see Figure 5(b)).

Synthesis of ne-scale details. For a given target frame in F T j , we now want to re-introduce the residual variations to the synthesized parametric transformation D. As illustrated in Figure 8, we deform the corresponding residual frames in f r 1 , . . . , f r K using D to produce a set of deformed example frames f r 1 . . . f r K . We then compute a set of pairwise pixel-level warping elds ϕ κ,κ : f r κ → f r κ ∀(κ, κ ) ∈ {1 . . . K } 2 using deformable image registration [START_REF] Glocker | Dense Image Registration Through MRFs And E cient Linear Programming[END_REF]]. Finally we apply a weighted n-way morph [START_REF] Lee | Polymorph: Morphing Among Multiple Images[END_REF]] to produce a single output frame f r by displacing and blending pixels in f r 1 . . . f r K according to the same weights w 1 . . . w K as in Section 5.2 and the warping elds ϕ κ,κ .

Sub-sequence stitching. Since the style transfer is applied independently on each animation sub-sequence, stylized overlapping subsequences need to be stitched together to avoid abrupt changes. We apply the same approach as described in previous paragraphs, but now only with two frames and with blending weights proportional to the temporal distance τ : to stitch overlapping sub-sequences i and i + 1, we use w i = τ /M i,i+1 and w i+1 = 1 -τ /M i,i+1 , where M i,i+1 is the number of frames in the transition.

Contact points adjustment. Our synthesis process does not guarantee that the resulting animation preserves the alignment with obstacles at contacts. This issue bears resemblance to the foot step detection/correction mechanism used when blending motions in skeletal animation. Yet our problem is simpler since we know the position of contact points; we can easily verify whether the spatial alignment with obstacles is preserved. If not, we simply shift or slightly rotate the synthesis result so that it aligns perfectly with the obstacle at collision time. To estimate the corresponding translation and rotation we use again image registration algorithm of [START_REF] Sýkora | As-Rigid-As-Possible Image Registration for Hand-Drawn Cartoon Animations[END_REF]. To avoid ambiguity in translation along the obstacle during the registration, we restrict the centroid of the synthesized drawing to move perpendicularly to (along the normal of) the nearest obstacle.

Texturing. We support two options to apply a texture inside the deformed drawings. The rst one takes as input a static image whose content is constant during the full sequence. This image is rst rotated according to the target sequence orientation, then it is registered with every residual frame f r i using [START_REF] Glocker | Dense Image Registration Through MRFs And E cient Linear Programming[END_REF]], and nally replaces those during the subsequent ne-scale synthesis steps (see Figure 13(c)). If the content of the texture varies in time, the artist needs to provide two versions of the style exemplar: one only showing the outline of the drawing and another with the full texture (see Figure 17). The former is used for quadratic registration whereas the latter is copied during the frame-by-frame synthesis.

Stroke appearance transfer. To o er additional artistic controls, we optionally allow the stroke appearance to be re-synthesized by exemplar using StyLit [START_REF] Fišer | StyLit: Illumination-guided Example-based Stylization of 3D Renderings[END_REF]. This can also help suppress resampling artifacts that may occur when applying the quadratic and free-form deformations. In practice we replace the complex illumination-based guidance with a simple gray-scale guiding channel G that softly demarcates positions of strokes in the source and in the target image (see Figure 9). 2016] -it takes as input an exemplar drawing containing few strokes created using the desired drawing media, e.g., color pencil (a), its corresponding guiding channel (b), the target frame synthesized using our method (c) and its guiding channel (d), and produces the resulting appearance transfer (e).

To generate G we lter the input stroke style exemplar as well as the output frame using the FDoG lter [START_REF] Kang | Coherent Line Drawing[END_REF]] which suppresses local discrepancies caused by physical properties of the artistic medium (in the stroke style exemplar) or by errors caused by warping and blending pixels (in the target stylized frame) and produces clean stroke outlines. Then a Gaussian blur is applied on this clean outline mask and the resulting bitmap image is normalized and used as a guiding channel for the StyLit algorithm. target sequence synthesis (one example trajectory) synthesis (all example trajectories) ground truth stylization Fig. 11. Results of our synthesis algorithm -(from le to right) target animation computed using physical simulation, synthesis using only one exemplar trajectory, synthesis using all available exemplar trajectories shown in Figure 10, ground truth stylization drawn by a professional artist.

RESULTS

We implemented our approach using a combination of Matlab/C++ code, and integrated it as a plug-in into "TVPaint Animation", a professional 2D animation software. In all our experiments, the input sequences were split into sub-sequences of N i = 2 3 #(e i-1 , e i+1 ) frames, where #(e i , e j ) is the number of frames between the key events e i and e j . The overlap M i,i+1 was set to 2 3 #(e i , e i+1 ). For source-target sub-sequence matching, N was set to 20 and the weights were set to λ vel = 1, λ dir = 1, λ rot = 50 to encourage consistent rotation.

To validate our method we designed two practical usage scenarios: (1) an o -line scenario where an existing database of pre-made style exemplars is used to perform stylization of new more complex sequences and (2) an interactive scenario where an animator speci es a set of sparse style exemplars in the target sequence and those are then used as a source for the synthesis of the entire sequence.

In the case of the o -line scenario, to create the database, we asked a professional animator to stylize a few physical simulations with a single ball (see Figure 10) and square bouncing on the ground (see Figure 1). This database of exemplars was then used to stylize more complex physical simulations containing multiple interacting objects in di erent environment (see Figures 1,11,12,16,13,17 and the supplementary video). For this scenario, we precompute parametric deformations for each exemplar in the database. Our implementation can precompute 10 frames in about 13 seconds on average. Using the precomputed deformations, synthesis of an animation containing 100 frames lasts approximately 56 seconds. The most time consuming parts of the pipeline are the image registration phases. Parallel processing could be used to accelerate the block matching step of [START_REF] Sýkora | As-Rigid-As-Possible Image Registration for Hand-Drawn Cartoon Animations[END_REF] and the graph-cut computation in [START_REF] Glocker | Dense Image Registration Through MRFs And E cient Linear Programming[END_REF]. In Figure 14 we also demonstrate the synthesized sequences with additional stroke appearance transfer which further enhances the hand-drawn look and feel.

For the interactive scenario, we let the artist work in TVPaint Animation, a professional animation software, using our plug-in to iteratively improve the stylization of a target sequence. The artist rst selects a few frames where a more pronounced stylization is required and draws over them. The target sequence is then resynthesized taking into account those modi ed frames, and the artist can inspect the result, tweak already modi ed frames, or provide stylization for additional frames (see the supplementary video for the recording of a live editing session). In this scenario, we compute parametric deformations only for newly stylized frames and then the synthesis is executed with comparable computational overhead as in the o -line scenario. In the captured session, the artist stylized only 5 percent of the frames until she was satis ed with the result.

The method could even work with a single stylized frame, but it will obviously produce visually pleasing results only when a few frames before and after a collision event are provided.

To con rm that our approach reproduces the hand-drawn style well, we asked the professional artist to stylize the entire sequence frame-by-frame (ground truth), and we also let our algorithm synthesize the same sequence using exemplars (1) from the database and (2) from a fraction of the stylized frames in the ground truth sequence (see Figure 11). Then, we asked two professional animators and three casual observers to compare the three results side-byside and assess the overall motion stylization quality. In most cases the results were almost indistinguishable from ground truth for non-professional observers. Professional artists were able to see some discrepancies namely in frames of which assumed stylization diverted signi cantly from the available exemplars, however, the overall response was very positive and they highly appreciate the advantage of having a trade-o between the nal visual quality and the time required to produce the exemplars.

In addition to the comparison with a ground truth sequence, we also provide an additional comparison showing the necessity of the decomposition into a quadratic deformation D and residual changes R. We synthesize sequences using only our parametric model and then compare with those synthesized using the full processing pipeline (see Figure 15).

LIMITATIONS AND FUTURE WORK

We have shown that our approach can produce good results using only a small subset of exemplar frames, and as a result can notably lower the amount of manual work required to create a variety of 2D animations. However, those are currently restricted to simple rigid bodies, and some practical -rather than conceptual -limitations need to be addressed to allow the stylization of more complex sequences. First, as mentioned in Section 4, for simplicity, our method does not explicitly model the spacing variations that an artist may introduce compared to the physical simulation (e.g., ease-in / ease-out e ects). This is especially noticeable in the dynamics of collisions. To a certain extent, those variations are implicitly captured by the translational part of the parametric transformation, but only as a linear frame-by-frame approximation when it should maybe be modeled as a continuous function along the rigid body trajectory. This can be seen in instances where a shape deforms along the contact plane just before or right after the collision (levitation e ect).

Besides, our approach takes advantage of rigid body symmetries which are helpful namely for the synthesis of rotations. In particular, exemplars of symmetric counterparts can be reused in situations where there is no corresponding sequence available in the original orientation. However, for asymmetric objects this simpli cation cannot be used and thus more exemplars need to be provided in order to be able synthesize consistent orientations. To further reduce the burden on the artist in such cases, it would be required to model the rotation of the example object separately and allow some amount of re-targeting during the transfer to the novel sequence.

The success of our method also strongly depends on the result of the automatic image registration phase, during which the quadratic deformation model is t to the stylized exemplar. For elongated objects such as the car in Figure 17, it would be interesting to explore the use of multiple quadratic deformation clusters such that the front of the object may be stylized earlier than its rear along the motion trajectory. More complex or excessive stylization may not t well to the available degrees of freedom and consequently the resulting residual changes may become overly large. This can cause di culties during the deformable image registration and the subsequent n-way morphing phase. As future work, we would like to explore di erent parametric models which would be able to describe a larger variety of deformations occurring in hand-drawn animations. Those more complex models will probably require a better image registration algorithm with a hierarchical processing that is able to adaptively add individual degrees of freedom to avoid getting stuck in some erroneous con gurations. It would also require a more advanced blending scheme.

When the exemplar animations di er considerably as compared to the target motion our method just picks a set of most similar ones which likely would not be appropriate. Moreover, when there is a large di erence in velocities, the dampening phase tends to remove the stylization e ect. Both e ects are undesirable and serve as a signal for the artist to provide some more appropriate exemplars.

A last drawback of our technique is that it requires multiple pixelbased warping operations to produce the nal shape. Although their number can be minimized by applying compound quadratic transformations, subsequent image re-sampling may still introduce artifacts that can alter the genuine appearance of the used artistic media. This limitation can be partially alleviated by additional postprocessing steps such as the proposed appearance re-synthesis or vectorization.

Combining all these improvements, we plan to extend our method to articulated objects such as cartoon characters. In this case a more complex, potentially hierarchical parametric model will be needed.

CONCLUSION

We have presented a method allowing to perform style transfer from an existing 2D hand-drawn animation exemplar to a more complex rigid body animation. To the best of our knowledge this is the rst attempt to provide a solution for such a challenging task. Despite the recent success of example-based stylization techniques focusing on appearance transfer, animation is still a mostly unexplored area and we believe that the results provided in this paper will motivate other researchers to explore and solve the large variety of challenges that emerges when considering transfer of motion style to more complex articulated objects such as cartoon characters.

  Fig. 1. Given a small set of exemplars consisting of computer-generated and hand-drawn 2D animation pairs (a), our method transfers to a new target sequence produced by physical simulation (b) both the high-level deformations and fine-scale appearance variations (c) present in the example animations. Optionally, the final appearance of the drawings can be modified by re-synthesizing di erent stroke textures (d).

  Fig. 2. Stylization analogy setup -given a set of frames F S coming from reference 2D rigid body source animations, corresponding hand-animated exemplars F E , and a new target animation F T , the synthesis algorithm relates physical parameters in F S and F T to produce the output stylized sequence F O that resembles F E . (Animations are depicted with onionskins, colored from green to blue according to frame numbers.)

Fig. 3 .

 3 Fig.3. Three scales hierarchical decomposition of the animation process, based on[Williams 2001, p.67].

Fig. 5 .

 5 Fig. 5. Deformation analysis -(a) The parametric deformation D is estimated using as-rigid-as-possible registration between the source f s and exemplar f e frames. (b) The residual frame f r is then computed by applying the inverse deformation D -1 on f e .

Fig. 6 .

 6 Fig.6. Visualization of the 10 modes defined by the quadratic deformation of[START_REF] Müller | Meshless Deformations Based on Shape Matching[END_REF].

Fig. 7 .

 7 Fig. 7. Blending quadratic deformations D -individual quadratic deformations D 1 , D 2 , D 3 estimated from the source frames f s 1 , f s 2 , f s 3 and their corresponding stylized counterparts f e 1 , f e 2 , f e 3 are blended together using the weights w 1 , w 2 , w 3 to obtain the resulting quadratic deformation D.

Fig. 8 .

 8 Fig.8. Synthesis of fine-scale details R -the synthesized quadratic deformation D is applied to individual residual frames f r 1 , f r 2 , f r 3 producing their deformed counterparts f r 1 , f r 2 , f r 3 . Those are then blended together using n-way morphing[START_REF] Lee | Polymorph: Morphing Among Multiple Images[END_REF]] to produce the resulting frame f r .

Fig. 9 .

 9 Fig. 9. Stroke appearance transfer based on StyLit algorithm[START_REF] Fišer | StyLit: Illumination-guided Example-based Stylization of 3D Renderings[END_REF]] -it takes as input an exemplar drawing containing few strokes created using the desired drawing media, e.g., color pencil (a), its corresponding guiding channel (b), the target frame synthesized using our method (c) and its guiding channel (d), and produces the resulting appearance transfer (e).

Fig. 10 .

 10 Fig. 10. Style pair database -overview of the source animations and their corresponding stylizations of a bouncing ball drawn by a professional artist.

  Fig. 12. Result of our synthesis algorithm for two bouncing balls -target sequence computed using physical simulation (le ); resulting synthesis computed using all available exemplar trajectories shown in Figure 10 (right).

  Fig. 13. Result of our synthesis algorithm for more colliding balls -(a) target sequence computed using physical simulation; (b) resulting synthesis computed using all available exemplar trajectories shown in Figure 10; (c) "stone wheel" texture applied during the frame-by-frame synthesis.

  Fig. 15. Comparison -synthesis using only the quadratic deformation model D (le ), and with the full processing pipeline which includes both the quadratic deformation D and the residual changes R (right). Note how the addition of the residual changes R significantly improves the expressiveness of the resulting animation.

  Fig. 16. Result of our synthesis algorithm for two interacting squares -the style pairs of Figure 1 are used, these only contain collisions with the ground plane.

ACM Transactions on Graphics, Vol. 36, No. 4, Article 127. Publication date: July 2017.

ACKNOWLEDGEMENTS

We would like to thank Zuzana Studená for preparation of handdrawn animations used in the paper, Petra Hapalová for integrating our algorithm into TVPaint, Vojtěch Votýpka for the recording of interactive sessions, and all anonymous reviewers for insightful comments and suggestions. This research was funded by Adobe and has been supported by the Technology Agency of the Czech Republic under research program TE01020415 (V3C -Visual Computing Competence Center) and by the Grant Agency of the Czech Technical University in Prague, grant No. SGS16/237/OHK3/3T/13 (Research of Modern Computer Graphics Methods).


Kevin Nadaud

Fabien Roubeau

Arnaud Pothier

Pierre Blondy
email: pierre.blondy@unilim.fr

Lin-Yang Zhang

Romain Stefanini

Compact Thin-Film Packaged RF-MEMS Switched Capacitors

Keywords: Microelectromechanical systems, Package, RF-MEMS, Switches

This paper presents the design, the realization and the measurement of a thin-film packaged RF-MEMS switched capacitors. Packaging is included in microelectronics fabrication process, with Silicon Nitride thin film. The capacitors are actuated by deflecting thin gold metal membranes towards the package dielectric, increasing the capacitance by a factor 2.3. The device size, including its packaging, is 50 × 40µm 2 . Pull-in and release voltages have less than 5V variation between 20°C and 85°C. The device has been tested with 20 dBm applied power and shows no sensitivity to incident power.

I. Introduction

Reconfigurable devices RF components are more and more needed to deal with the ever-increasing demand for wireless data access. With the growing number of telecommunication standards, reconfigurable and tunable front ends are needed to limit the complexity of radio front ends. Switched RF-MEMS capacitors are among the serious candidates to realize this function thanks to their low losses, high linearity and very low power consumption compared to semiconductors [START_REF] Rebeiz | RF MEMS Theory, Design and Technology[END_REF].

Single RF MEMS devices can be combined into pseudo digital arrays of switched capacitors, with combinations resulting in well-defined step variations with high linearity and low loss. Other technologies, like series combination of stacked gate MOS transistors switches with fixed capacitors can be used for the same purpose. These semiconductor technologies offer moderate Q s , but they can be made at very low cost, and are very easy to integrate in PC-board RF-circuits.

Indeed, considerations like cost, size, and easiness of integration are critical to the success of RF-MEMS technology. In particular, specific MEMS packaging has been one of the main roadblock against rapid and successful adoption of this technology.

Hermetic wafer-to-wafer bonding with glass-frit intermediate bond layer has successfully been used on commercial switches, but the thickness and profile of the resulting components make them difficult to integrate as surface-mounted elements. nology. There have been several efforts in the past towards the development of low cost, low profile thin film packaging. The idea is to use thin film processing steps for building dielectric shells above micro-electromechanical parts. Most proposed concepts in RF-MEMS stem from early work presented in [START_REF] Leedy | Thin-film encapsulated RF MEMS switches[END_REF], and have been further developed in [START_REF] Wunnicke | Small, low-ohmic RF MEMS switches with thin-film package[END_REF]- [START_REF]kinetics[END_REF], on various ohmic contact and capacitive RF-MEMS structures.

The structures presented in this paper are using miniature RF-MEMS capacitive structures [START_REF] Mercier | Miniature RF MEMS switched capacitors[END_REF]- [START_REF] Sedaghat-Pisheh | RF MEMS miniature-switched capacitors with pull-down and pull-up electrodes for high power applications[END_REF], integrated into silicon nitride dielectric shells.

The compactness of the proposed components reduces sensitivity to temperature variations and will permits the integration of compact switched capacitors arrays.

II. Fabrication

A cross section of the proposed switched capacitor is shown in Fig. 1 along with a sketch of a conventional capacitive switch. In the proposed design, the RF signal electrode is standing above the deflectable metal membrane, and the dielectric layer is also used as a protective shell for the structure.

The processing steps are shown in Fig. 2. Switched capacitors have been fabricated on 400 µm thick silicon substrate, on which a 500 nm thick thermally grown SiO 2 on both sides.

Processing starts with the deposition and patterning of a first 300 nm thick sacrificial layer (Fig. 2a). Next, a 5 nm/400 nm Ti/Au membrane metal, with tensile stress, is evaporated and patterned. Then a second 300 nm thick sacrificial layer is deposited and patterned (Fig. 2b andc).

A 500 nm thick Si 3 N 4 layer is then deposited on top of the previous layers and the good conformal properties of the PECVD deposition techniques allowing maintaining good rigidity of the structure (Fig. 2d).

Small holes have been etched on the side of the dielectric shells to permit the release of the structure. After the sacrificial layers have been etched away, the devices are dried in a critical point drying system.

The last metal layer is 1.5 µm thick low-stress Al layer. This final layer closes release holes and thus finalizes the process of the switch. This metal layer also allows us to realize the top electrode and the CPW lines used for the switch (Fig. 2f). The volume of the fabricated cavities is about 1.5 pL, which is extremely small compared to other packaging techniques for RF-MEMS.

The switch dimensions are given in Fig. 3. The active area is only 40×50 µm 2 , including package. The moveable beam is 40 µm long by 30 µm wide. FEM and analytical [START_REF] Rebeiz | RF MEMS Theory, Design and Technology[END_REF] computed characteristics are summarized in Table I.

The devices have been simulated using ADS, and an equivalent scheme has been fitted to measurements.

III. Measurements

The measured S -parameters are shown in Fig. 4 along with equivalent scheme modeling. No de-embedding has been performed on the presented measurements, and the presented results include aluminum pads to RF probe loss, and access pads substrate losses. The silicon resistivity adds a significant amount of loss at the access, and this phenomenon has been fitted using conventional CMOS access model.

The probe to contact pads resistance could not be completely de-embedded and the actual series resistance of the switched capacitance was difficult to extract. Therefore, the capacitor Q could not be extracted accurately. However, the 1.5 µm thick metal Aluminum top layer strongly reduces ohmic loss compared to previously published designs [START_REF] Mercier | Miniature RF MEMS switched capacitors[END_REF]- [START_REF] Sedaghat-Pisheh | RF MEMS miniature-switched capacitors with pull-down and pull-up electrodes for high power applications[END_REF]. Also, because of the small size of the device, no inductor in series was needed to model the capacitor up to 40 GHz (Fig. 5).

The devices have been biased using 1 kHz bipolar square waveform. When the bias voltage is applied, the extracted capacitance is 53 fF and when the bias voltage is removed, the capacitance value goes back to 23 fF, which is in good agreement with ADS Momentum simulations (Table I, 20 fF). The corresponding contrast is 2.3. The on state capacitance could not be predicted accurately by EM simulation because of the roughness of the dielectric layer.

The measured pull-in voltage is 25 V, in good agreement with analytical computation [START_REF] Rebeiz | RF MEMS Theory, Design and Technology[END_REF] (Table I, 29 V), and the release voltage is 17 V.

The pull-in and release voltages have been measured has function of the temperature and are reported Fig. 6. Because of the small size of the device, the temperature dependence is quite low between 20 and 85 • C. It is noteworthy that the temperature dependence comes both from the shell and the beam deformations.

The influence of the incident power is shown Fig. 7 for the two states. One can note the power does not lead to a variation of the transmission coefficient, showing that the capacitance is insensitive to the power up to 20 dBm. In the present configuration, the capacitance is in series and the incident power generates maximum voltage across the device.

IV. Conclusion

A novel switched capacitor has been designed, fabricated and tested. Extremely small packaging has been integrated in the fabrication process, using standard microelectronics processing steps. The capacitor can be switched between 23 and 53 fF, and can be easily put into arrays of devices in order to obtain compact digital-like variable capacitors. The measured capacitance contrast is 2.3 but it can be easily increased by adding a third electrode to obtain a tri-plate configuration as proposed in [START_REF] Sedaghat-Pisheh | RF MEMS miniature-switched capacitors with pull-down and pull-up electrodes for high power applications[END_REF]. The devices are using relatively thick metals, and will permit the fabrication of very low loss, zero level packaged RF-MEMS variable capacitors arrays.

Figure 2 .

 2 Figure 2. Fabrication steps of the switched capacitor.

Figure 3 .

 3 Figure 3. Top view (a) and SEM (b) picture of a fabricated switched capacitor. The released capacitor cannot be seen on the SEM picture since it is covered by the dielectric shell.

Figure 4 .Figure 5 .

 45 Figure 4. Measured (solid line) and equivalent scheme (dashed line) responses.

Figure 6 .Figure 7 .

 67 Figure 6. Pull-in voltage V p and release voltage V r as a function of temperature.

Table I Switched

 I Capacitor Characteristics.

	Beam length l	40 µm
	Beam width w	30 µm
	Pull-in voltage V

p 25 V (Meas.) / 29 V (Sim.) Release voltage V r 17 V (Meas.) Mechanical resonance frequency 1.01 MHz (Sim.) Switching times @ 1.5V p 384 ns (Sim.) On-state capacitance 53 fF (Meas.) Off-state capacitance 23 fF (Meas.) / 20 fF (Sim.)

Acknowledgment

The authors would like to acknowledge the financial support of the European Space Agency, ESA-ESTEC, Noordwijk, The Netherlands and Airbus Defence and Space, Saint Medard en Jalles, France.


Kevin Nadaud

Fabien Roubeau

Arnaud Pothier

Pierre Blondy
email: pierre.blondy@unilim.fr

Lin-Yang Zhang

Romain Stefanini

High Q Zero Level Packaged RF-MEMS Switched Capacitor Arrays

Keywords: Microelectromechanical systems, Package, RF-MEMS, High-Q

This paper presents the design, the realization and the measurement of a thin-film packaged RF-MEMS switched capacitors. Packaging is included in microelectronics fabrication process, with Silicon Nitride thin film. The capacitors are actuated by deflecting thin gold metal membrane towards the package dielectric, increasing the capacitance by a factor 2.47. The device size, including its packaging, is 80 × 90µm 2 . The highest quality factor of a single capacitor is 1515 at 5 GHz, which comes from the low parasitic capacitance and the thick metallization. To obtain larger capacitance value, an array of 4 × 2 capacitors has been realized and measured.

I. Introduction

Due to the ever-increasing demand for wireless data access, reconfigurable devices RF components are more and more needed. With the growing number of telecommunication standards, reconfigurable and tunable front ends are needed to limit the complexity of radio front ends. Switched RF-MEMS capacitors are among the serious candidates to realize this function thanks to their low losses, high linearity and very low power consumption compared to semiconductors [START_REF] Rebeiz | RF MEMS Theory, Design and Technology[END_REF].

Single RF MEMS devices can be combined into pseudo digital arrays of switched capacitors, with combinations resulting in well-defined step variations with high linearity and low loss. Other technologies, like series combination of stacked gate MOS transistors switches with fixed capacitors can be used for the same purpose. These semiconductor technologies offer moderate Q s , but they can be made at very low cost, and are very easy to integrate in PC-board RF-circuits. Indeed, considerations like cost, size, and easiness of integration are critical to the success of RF-MEMS technology. In particular, specific MEMS packaging has been one of the main roadblock against rapid and successful adoption of this technology.

Hermetic wafer-to-wafer bonding with glass-frit intermediate bond layer has successfully been used on commercial switches, but the thickness and profile of the resulting components make them difficult to integrate as surface-mounted elements. Moreover, these packaging techniques are very specific to MEMS and are difficult to insert in low cost CMOS fabrication process flow [2], [3]. Low processing effort and MEMS-unspecific packaging techniques are of high interest for the progress of this technology. There have been several efforts in the past towards the development of low cost, low profile thin film packaging. The idea is to use thin film processing steps for building dielectric shells above micro-electromechanical parts. Most proposed concepts in RF-MEMS stem from early work presented in [START_REF] Leedy | Thin-film encapsulated RF MEMS switches[END_REF], [START_REF] Nadaud | Compact thin-film packaged RF-MEMS switched capacitors[END_REF], and have been further developed in [START_REF] Wunnicke | Small, low-ohmic RF MEMS switches with thin-film package[END_REF]- [START_REF]kinetics[END_REF], on various ohmic contact and capacitive RF-MEMS structures.

The structures presented in this paper are using miniature RF-MEMS capacitive structures [START_REF] Mercier | Miniature RF MEMS switched capacitors[END_REF]- [START_REF] Sedaghat-Pisheh | RF MEMS miniature-switched capacitors with pull-down and pull-up electrodes for high power applications[END_REF], integrated into silicon nitride dielectric shells. The compactness of the proposed components reduces sensitivity to temperature variations and will permits the integration of compact switched capacitors arrays, which permits to obtain large capacitor values or digitally tunable capacitors (DTC).

II. Fabrication

A cross section of the proposed switched capacitor is shown in Fig. 1 along with a sketch of a conventional capacitive switch. In the proposed design, the RF signal electrode is standing above the deflectable metal membrane, and the dielectric layer is also used as a protective shell for the structure.

The processing steps are shown in Fig. 2. Switched capacitors have been fabricated on 400 µm thick silicon substrate, on which a 500 nm thick thermally grown SiO 2 on both sides.

Processing starts with the deposition and patterning of a first 300 nm thick sacrificial layer (Fig. 2a). Next, a 5 nm/400 nm Ti/Au membrane metal, with tensile stress, is evaporated and patterned. Then a second 300 nm thick sacrificial layer is deposited and patterned (Fig. 2b andc). A 500 nm thick Si 3 N 4 layer is then deposited on top of the previous layers and the good conformal properties of the PECVD deposition techniques allowing maintaining good rigidity of the structure (Fig. 2d). Small holes have been etched on the side of the dielectric shells to permit the release of the structure. The last metal layer consists of a low-stress gold layer. Firstly, a 5 nm/200 nm Ti/Au seed layer is deposited by evaporation, then 1 µm of gold is electroplated in order to obtain thicker metallization. This metal layer also allows us to realize the top electrode and the CPW lines used for the switch (Fig. 2e). Finally, the sacrificial layers have been etched away, the devices are dried in a critical point drying system. The volume of the fabricated cavities is about 1.5 pL, which is extremely small compared to other packaging techniques for RF-MEMS.

The switch dimensions are given in Fig. 3. The active area is only 75 × 80 µm 2 , including package. The moveable beam is 55 µm long by 30 µm wide. FEM and analytical [START_REF] Rebeiz | RF MEMS Theory, Design and Technology[END_REF] computed characteristics are summarized in Table I.

The devices have been simulated using ADS, and an equivalent scheme (Fig. 4) has been fitted to measurements. The silicon resistivity adds a significant amount of loss at the access, and this phenomenon has been fitted using conventional CMOS access model.

III. Measurements

A. Single capacitor

The measured S -parameters are shown in Fig. 5 along with equivalent scheme modelling. No de-embedding has been performed on the presented measurements, and the presented results include gold pads to RF probe loss, and access pads substrate losses.

The devices have been biased using 1 kHz bipolar square waveform. When the bias voltage is applied, the extracted capacitance is 52 fF and when the bias voltage is removed, the capacitance value goes back to 21 fF, which is in good agreement with ADS Momentum simulations (Table I, 20 fF). The corresponding contrast is 2.47. The on state capacitance could not be predicted accurately by EM simulation because of the roughness of the dielectric layer. The series resistance R s of the capacitor switch is quit hard to extract since the major part of the losses comes from the probe-pads but the value is less than 1 Ω. This permits to computed the quality factor with the following definition:

Q = 1 R s C MEMS ω (1)
This gives a quality factor at 5 GHz, of 1515 for the unbiased state and of 612 for the biased state. This high Q comes from the relatively thick metallization (400 nm for the beam and 1.2 µm for the top metal). Moreover, the the very low parasitic capacitance (around 6 fF), due to the structure of the MEMS, permits to conserve significant tunability factor with low capacitance value and obtain a high Q.

The measured pull-in voltage is 25 V, in good agreement with analytical computation [START_REF] Rebeiz | RF MEMS Theory, Design and Technology[END_REF] (Table I, 29 V), and the release voltage is 17 V.

B. Capacitor array

A single packaged MEMS capacitor having a small capacitance value, a capacitor array has been design with 8 unit elements assembled in parallel (Fig. 6). Each capacitor has the same characteristics than the single capacitor describes previously. The equivalent modelling scheme is the same for the array since the connexion lines are quite small and do not add parasite inductance in the considered frequency range. The equivalent capacitor of the array is 400 fF in the on-state and 173 fF in the off-state which corresponds to 8 times the capacitance of the single capacitor. The tunability factor of the array is 2.3 which is slightly lower than the capacitor one which may comes to the parasitic capacitances between beam and lines. Thanks to relatively thick top metallization made of gold, the equivalent series resistance stay very low (< 1.5 Ω).

IV. Conclusion

A novel switched capacitor has been designed, fabricated and tested. Extremely small packaging has been integrated in the fabrication process, using standard microelectronics processing steps. The capacitor can be switched between 21 and 52 fF, and can be put into arrays of devices in order to obtain compact digital-like variable capacitors. The measured capacitance contrast is 2.47 but it can be easily increased by adding a third electrode to obtain a tri-plate configuration as proposed in [START_REF] Sedaghat-Pisheh | RF MEMS miniature-switched capacitors with pull-down and pull-up electrodes for high power applications[END_REF]. The measured quality factor at 5 GHz of the single capacitor is respectively 1515 and 612 for the unbiased and the biased states, which is one order of magnitude higher than semiconductor technologies. The single capacitor has been successfully integrated into a 4 × 2 shunt capacitor array which permits to obtain a larger values of the capacitance and paves the way to digitally tunable capacitor.

Figure 1 .

 1 Figure 1. Operating principle of a conventional RF-MEMS capacitive switch (a) and proposed switched capacitor (b).

Figure 2 .

 2 Figure 2. Fabrication steps of the switched capacitor.

Figure 3 .

 3 Figure 3. Top view (a) and microphotography (b) of a fabricated switched capacitor.

Figure 5 .

 5 Figure 5. Measured (solid line) and equivalent scheme (dashed line) responses of the single capacitor. C ox = 0.5 pF, R sub = 7 kΩ and R s = 1 Ω. C MEMS is 52 fF in the on-state and 21 fF in the off-state.

Figure 6 .Figure 7 .

 67 Figure 6. Optical photography of the capacitor array.

Table I .

 I Switched Capacitor Characteristics.Figure 4. Equivalent scheme of the switched capacitor.

	Beam length l	55 µm
	Beam width w	30 µm
	Pull-in voltage V p	25 V (Meas.) / 29 V (Sim.)
	Release voltage V r	17 V (Meas.)
	Mechanical resonance frequency	1.01 MHz (Sim.)
	Switching times @ 1.5V p	384 ns (Sim.)
	On-state capacitance	52 fF (Meas.)
	Off-state capacitance	21 fF (Meas.) / 20 fF (Sim.)
	C MEMS	R s
	C ox	C ox
	R sub	R sub