Abdallah Karakra
email: abdallah.karakra@mines-albi.fr

Franck Fontanili
email: franck.fontanili@mines-albi.fr

Elyes Lamine
email: elyes.lamine@mines-albi.fr

Jacques Lamothe
email: jacques.lamothe@mines-albi.fr

HospiT'Win: A Predictive Simulation-Based Digital Twin for Patients Pathways in Hospital

Keywords: Digital Twin, Hospital of the Future, Internet of Things, Modeling, Simulation

Hospitals are starting to move away from traditional based-systems to the information technology based-systems. Today, Internet of Things (IoT), Body Sensor Network (BSN), Modeling, Simulation, and Artificial Intelligence (AI) are core technology elements that will be used in hospital of the future to improve the quality of patient care. Collecting the patient's data and monitoring their states and behavior became mandatory to improve their care. This paper proposes a novel framework for supporting the hospital of the future named HospiT'Win. This framework uses the core technology elements mentioned above to create a digital twin, that is a virtual replica of the hospital, allowing the health care providers to trace the patient's pathways data, monitor their behaviors, and predict their near future outcomes. So that, they can provide the right care in a right place, and in a right time. The paper explains in details the main components, the structure, and the way to synchronize the state and the behavior of the digital twin with the patients pathways in the real hospital. In case of unexpected events, HospiT'Win predicts the near future to see their impact on the real hospital. Moreover, it provides the possible solutions to minimize the impact of these events to preserve the quality of health care inside the hospital.

I. INTRODUCTION

Digital twin (DT) is attracting a lot of attention and hype in the recent years, and is being considered as a key research area. There are many scientific research papers that showed the importance of applying the digital twin in the modern industry [START_REF] Negri | A review of the roles of digital twin in cps-based production systems[END_REF]. At the best of our knowledge, no scientific research paper showed the importance of applying the digital twin in healthcare organizations for managing the patients pathways in hospitals. In scientific research, several definitions for the digital twin are existing, but there is no clear, specific and unique definition for this term [START_REF] Negri | A review of the roles of digital twin in cps-based production systems[END_REF] [START_REF] Kritzinger | Digital twin in manufacturing: A categorical literature review and classification[END_REF]. Most of the existing definitions relate to the life of a vehicle [START_REF] Shafto | Modeling, simulation, information technology & processing roadmap[END_REF], future of manufacturing processes, and product life cycle management (PLM). This paper discusses in details its application in hospital, called HospiT'Win.

Digital twin is not exactly new. Actually, the core elements of a digital twin are already well known and exist such as modeling, simulation and internet of things (IoT). The idea behind the digital twin can be summarized as follows: there is (1) a real world, (2) a virtual world, (3) a link of data from the real world to the virtual world and (4) a link of information and knowledge from the virtual world to the real world as illustrated in Fig. 1. In particular, the following questions are developed: What is a digital twin for patients pathways in hospital? What are the levels and the models that are used in the creation of HospiT'Win? How to link the HospiT'Win with the real hospital so that they can synchronize up-to-date data and information? How the HospiT'Win can predict the impact of unexpected events in near future? How can minimize the effect of this impact to preserve the quality of care from any future risks or problems?

This paper is organized as follows: literature review is provided in section II. Then the proposed approach is explained in section III and the conclusion in section IV.

II. LITERATURE REVIEW

The original idea behind the "twin" returned back to the NASA's Apollo project (National Aeronautics and Space Administrations), when NASA created two exact copies of space vehicles. The one that was on the earth was referred to as "twin", utilized to mirror the condition of the space vehicle during the mission [1] [4]. The concept of the Digital Twin has been used in Product Lifecycle Management (PLM), in a previous work at the University of Michigan in 2002 in a presentation for Grieves [START_REF] Karakra | Pervasive computing integrated discrete event simulation for a hospital digital twin[END_REF]. Today, in scientific literature, there are many explanations and definitions for the digital twin. Those definitions are related to the life of a vehicle, future of manufacture, and product lifecycle management (PLM).

First definitions of digital twin came for a vehicle and have been extended to products: B.T. Gockel et al. [START_REF] Gockel | Challenges with structural life forecasting using realistic mission profiles[END_REF] define the digital twin as "Ultra-realistic, cradle-to-grave computer model of an aircraft structure that is used to assess the aircrafts ability to meet mission requirements". For K. Reifsnider et al. [START_REF] Reifsnider | Multiphysics stimulated simulation digital twin methods for fleet management[END_REF], the digital twin is "Ultra-high fidelity physical models of the materials and structures that control the life of a vehicle". For Y. Bazilevs et al [START_REF] Bazilevs | Isogeometric fatigue damage prediction in large-scale composite structures driven by dynamic sensor data[END_REF], the digital twin is "High-fidelity structural model that incorporates fatigue damage and presents a fairly complete digital counterpart of the actual structural system of interest". Other definitions and explanations for the digital twin are in [START_REF] Gabor | A simulation-based architecture for smart cyber-physical systems[END_REF], [START_REF] Kraft | The air force digital thread/digital twin-life cycle integration and use of computational and experimental knowledge[END_REF], [START_REF] Canedo | Industrial iot lifecycle via digital twins[END_REF], [START_REF] Schluse | From simulation to experimentable digital twins: simulation-based development and operation of complex technical systems[END_REF].

The most common definition used today for the digital twin of a product is given by Glaessgen, Stargel (2012) "digital twin is an integrated multi-physics, multi-scale, probabilistic simulation of a complex product and uses the best available physical models, sensor updates, etc., to mirror the life of its corresponding twin." [START_REF] Tao | Digital twindriven product design, manufacturing and service with big data[END_REF].

In short, there are many definitions in the scientific literature for the digital twin. Most of these definitions agree that the digital twin should have the following core elements: Model, Simulation, Sensors, and IoT. This paper exploits those core elements to describe our proposed approach.

III. PROPOSED APPROACH

This section is divided into two parts. The first part illustrates in details the framework of HospiT'Win. The second part proposes a way to link and to synchronize the HospiT'Win with real hospital in the near-real time.

A. HospiT'Win: a digital twin for patients pathways in hospital

HospiT'Win is a new class of digital twin. It proposes to support the patient pathways in hospital of the future. It is composed of three levels of functionalities: construction and validation level, tracking and monitoring level, and vision of the future level as illustrated in Fig. 2.

The three levels presented in Fig. 2 are differing in the way of data integration between the real sensor objects and the digital model. At the validation level, a meta-model is configured manually and not connected or synchronized with any data or real sensor objects directly. This type of connection can be called "offline" connection. The model is manually fed by data extracted from hospital databases. It enables to specify the perimeter of the digital twin, list of concerned resources, granularity and precision of the models, associate sensors to events. In level two, the discrete event simulation model (DESmodel) is connected and synchronized to work in the near-real time with real sensor objects. This connection can be called "online" connection. Data are captured by the sensors to feed a synchronized model. The predictive model in the third level is not synchronized to the real world. It is used to simulate the near future in case of unexpected events from the current state of the real world. This state is based on the flow of data received from the database that exists at the second level.

For more clarification, this paper would like to discuss each level separately. It will discuss in details the models, the type of connections and the techniques utilized in each level.

Construction and Validation level: At this level, there is no actual connection or synchronization between the virtual world (virtual hospital) and the real world (real hospital). In other words, any change in the state of the real hospital have no effect on the virtual hospital and vice versa. Fig. 3 shows the main parts of this level.

Fig. 3. Construction and validation level

This level is composed of real sensor objects in the real world, digital meta-models in the virtual world, and the database contains data and information about each real sensor object in the real world.

Real sensor objects is any entity inside the real hospital that has a sensing capability, so that the sensors can capture/collect near-real time data about this entity.

In this paper, the digital meta-model is defined as a virtual representation of the real sensor objects with a set of concepts. It is typically used to describe what a real sensor object will look like, and what are the states and behaviors of this object. It just describes the parts and the components of the real sensor object and how they interact with each other. There are no automated data or information exchange between the meta-model and the real sensor object. The database is the only technique used to feed the meta-model with data and information about the real sensor object.

The main function for this level is to construct a metamodel for the real sensor objects as initial developments and to validate if the models are correctly implemented to have the same state and behavior as the real sensor objects. After that, the meta-model is used to build the Discrete Event Simulation living models in the second level.

Tracking and monitoring level: Fig. 4 illustrates a detail picture of the second level (tracking and monitoring level). This online level is composed of real sensor objects, IoT devices (including wireless sensor networks (WSNs), radiofrequency identification (RFID)) and multi-processing layers (including preparing and cleaning data layer, security layer, and dynamic database) to synchronize the virtual world (DESmodel) to the real world. The DES-model is a digital living Fig. 4. Tracking and Monitoring level model, in which the data flow in one way from the real sensor object to the digital object based on a certain event. This model reflects the behavior and the state of the real sensor object in the near-real time. Any updates and changes on the state and behavior of the real sensor object leads to update the state and behavior of the digital object in the near-real time. This level contains three processing layers to prepare the data before transferring from the real sensor objects to the digital models as follows:

1) Data preparing and cleaning layer: the responsibility of this layer is to clean and to prepare the data before sending to the DES-model. Actually, the data provided by the sensors are potentially unreliable and need to be cleaned before their use [START_REF] Jeffery | Declarative support for sensor data cleaning[END_REF], [START_REF] Zhuang | A weighted moving average-based approach for cleaning sensor data[END_REF]. 2) Security layer: the responsibility of this layer is to ensure that the data transfer from the real sensor objects to the DES-models are secure [START_REF] Wang | Dependable and secure sensor data storage with dynamic integrity assurance[END_REF]. 3) Dynamic Database layer: to store the updated data and information. DES-model uses the database to monitor and to track the flow of data from the real sensor objects to the digital models. It stores the feedback about the behavior of the models to confirm that it correctly works with respect to the real hospitals KPIs (Key Performance Indicators) which make easier to detect the risk and fix before occurring. The main purpose for this level are summarized as follows:

• Uses for near-real time monitoring (like a live video):

it provides information about the near-real time state and behavior of the real sensor objects (flow of patients, operating room monitoring, registration desk, etc.). • Tracks the flow of data from the real world to the virtual world.

• Detects the unexpected situation. For example, before the problem occurs it generates an event to call the predictive model to see the future (delay, change in schedule,.. etc). Vision of the future level: This level is composed of predictive model, stochastic simulation, and artificial intelligence (AI) techniques as demonstrated in Fig. 5. In this Fig. 5. Vision of the Future Level paper, predictive model is a digital representation that applies stochastic simulation techniques on the near real time data and information that receives from the level two database. In case of unexpected events, it predicts the near future outcomes to see the impact of these events in the real hospital. Also, it generates various future scenarios to minimize the impact of these events. It uses artificial intelligence techniques to determine the acceptable scenario to adopt in the real hospital with many details about the risks, financial, etc. The following steps shows the main functions for this level.

1) In case of unexpected issues in level two, unexpected event is generated. 2) Level two triggers the generated event with details about the issues.

3) The predictive digital model handles the event by applying the stochastic simulation on the event's data to predict the near future with possibility to generate scenarios to minimize or to solve the impact of these issues. 4) AI techniques use to select the best applicable scenario to minimize the impact of these issues in the real hospital. 5) Before applying the selected scenario in the real hospital, connection is opened with the decision makers. Decision makers take their decision based on the available knowledge and performance of the selected scenario. 6) Decision makers can contact external partners (pharmacies, ambulances, other hospitals, etc.) to be confident with a decision that is proposed. 7) Based on the confident percentage the decision could apply or not.

B. Linkage between the virtual hospital and the real hospital

The paper proposes to use a database to synchronize the virtual hospital with the real hospital in a near-real time and based on discrete events (DE).

The following steps summarize the way to link and synchronize the virtual hospital with the real one.

1) Create a database that contains the following types of tables:

• Tables to store data, information, state, behavior, etc. about each real sensor object (patients, doctors, etc.). • Tables to store feedback that come from the DESmodels. • Tables for hospital KPIs (length of stay, staff satisfaction, patients satisfaction, etc.). 2) Thanks to the WSNs, and RFIDs, data are collected about the hospital real sensor objects and send to the database over the internet. 3) Updated data from the database are pushed to and read by the virtual hospital (open connection to the database). 4) The feedback data transfer from the virtual hospital to the database (feedback tables updated). Those technologies could be used to make the connection: MySQL database, ODBC driver, and Windows Data Source.

IV. CONCLUSION

This paper proposed a novel framework to support the hospital of the future named HospiT'Win. In case of unexpected events in the real hospital, this framework can predict the near future to see what will be the impact on the hospital and what are the possible solutions to reduce this impact. Moreover, the paper demonstrated a way to connect the HospiT'Win with real hospital; This allows to track, monitor and validate that the hospital functionality goes in the right direction and in a right time.

Fig. 1 .

 1 Fig. 1. Digital Twin This paper aims to propose a new class of digital twin to support patient pathways in a hospital named HospiT'Win.In particular, the following questions are developed: What is a digital twin for patients pathways in hospital? What are the levels and the models that are used in the creation of HospiT'Win? How to link the HospiT'Win with the real hospital so that they can synchronize up-to-date data and information? How the HospiT'Win can predict the impact of unexpected events in near future? How can minimize the effect of this impact to preserve the quality of care from any future risks or problems?This paper is organized as follows: literature review is provided in section II. Then the proposed approach is explained

Fig. 2 .

 2 Fig. 2. HospiT'Win Levels


Architects on the Edge

The French architecture critic and consultant Valéry Didelon says it is time for architects to reclaim the border territories and invade suburbia -but under new terms of engagement.

In today's world where urbanisation reigns supreme, the town centre has paradoxically slipped from the spot-light. Around the globe, people, wealth and knowledge are concentrated in major urban areas -metropolisation -but their forms and functions are spreading ever further into suburbia -suburbanisation. This process of centralisation and decentralisation began in the U.S. in the 1920s before advancing across Europe in the 1950s and 60s and then continuing to spread worldwide. New York City now has only 30 percent of the population in its inner city, Paris barely over 20 percent and the cities of Germany's Ruhr region under 6 percent. [START_REF] Moriconi-Ebrard | De Babylone à Tokyo[END_REF] The traditional city as well as pristine natural areas, have become isolated islands in a sea of suburbia ex-tending to the horizon. In 1964, Melvin M. Webber dubbed this phenomenon the "Nonplace Urban Realm".

Although the edge condition is increasingly documented and debated these days, the architect's place in the blueprint is little discussed, probably because it is so limited. Suburban structures seem better described as construction than as architecture. Quality is poor, and few spend the resources to actually build. Buildings are often purely functional, designed with little care and lacking durability. Residential developments set themselves apart by their mediocrity and homogeneity, factories and offices are blindingly unimpressive and commercial buildings are borderline shoddy structures. One common thread in the suburban development process is that it all generally occurs without input from architects, who have been relegated to the sidelines. However, this was not always the case. After World War II, the leaders of the modernist movement had their chance. Areas on the fringe were a blank canvas for their experiments, as illustrated by France's massive housing projects. This was but a brief moment of glory, as architects fled the suburbs amidst jeers, victims of their own arrogance. In response, they abandoned these frontiers. In the 1970s and 80s, as suburbanisation spread in Europe, architects championed the postmodern retour à la ville movement in Italy, France and Belgium. They were so unwilling to acknowledge the spread of progress to the suburbs that it escaped them altogether. Instead, they seemed to link their fate to that of the historic city, and embellishing and improving city centres and the surrounding neighborhoods became the only legitimate focus of their profession. The architect's territory was thus limited to the "downtown area" where even today, most continue to base their offices.

But here at the start of the 21st century it is no longer possible to ignore the issue of the suburbia as it infringes upon the historic city. City-dwellers' lifestyles are increasingly similar to those of suburban residents, peripheral architecture is creeping toward city centres and suburban developments are becoming the model for shaping downtown areas of old. With each passing day, architects are more confronted with the edge condition.

The triumph of suburbia, the fall of the traditional city, and the decline of the traditional values of order and harmony arise, along with a major crisis in the architectural profession.

What position should architects take with regard to this interloper that has relegated them to the sidelines and challenged their legitimacy? Upon first glance, it seems that they must choose between two evils: One is to fight against suburbia, holding tight to their values and even counter-attacking, with the movement to "urbanise suburbia", as has been recently heard in France. [START_REF]During the unrest in France in autumn 2005, architects and politicians agreed on the necessity of "civilizing the youth" and "urbanizing suburbia[END_REF] However, in doing so they miss what is going on in the outskirts, they stay detached from their development. The second is to embrace suburbia, incorporating its mode of propagation, accommodating its aesthetic and participating in its expansion, but with this choice they run the risk of watering down their art and struggling to justify their contribution. Will architects be forced to choose between resisting and fading into extinction or acquiescing and perishing in the sea of banality?

Not necessarily, if we take a historical look at how they were able to conquer worlds that escaped them. At the end of the 19th century, development of big-city industrial landscapes -coal yards and steel mills, ports and ware-houses -was presided over by engineers. These filthy realms were eschewed by architects, who focused instead on embellishing bourgeois urban centres. Architects snubbed this new reality and buried their heads in the sand of the declining Beaux-Arts culture. The revival was thus led by avant-garde thinkers who took interest in the urban fringe. In 1913, the pioneer Walter Gropius made use of grain silos seen in North America, [START_REF] Gropius | Die Entwicklung moderner Industrie-baukunst[END_REF] and was imitated by Le Corbusier in 1923 in Vers une architecture. Following in their footsteps, many modern architects undertook to outfit raw industrial areas in the fabrics of their art. In the late 1960s, a similar process occurred when intellectuals like Reyner Banham and architects like Denise Scott Brown and Robert Venturi left the beaten path to explore the commercial realm so disdained by their contemporaries. [START_REF] Banham | Los Angeles: The Architecture of Four Ecologies[END_REF] Here, they discovered a "low culture" full of vitality with the ability to breathe new life into their aging "high culture". They learned from this vernacular landscape, not reproducing it, but rather introducing it into the academic culture. By reconnecting with the city as it is, rather than how it should be, they wanted to restore the legitimacy compromised by the profession's elitist attitude. Like their predecessors, it is only by venturing outside of the city and away from the architectural culture that the Venturis managed to strengthen their art.

However, none of these innovators actually transformed iron into gold. Although they may have brought new vitality to their art, they did not completely succeed in reforming their profession and never gained a permanent foothold in these areas. They remain foreigners to the edge condition. They are spectators rather than participants and seem resigned to their impotence, as illustrated by Rem Koolhaas' statement: "Urbanism will not only, or mostly, be a profession, but a way of thinking, an ideology: to accept what exists. We were making sand castles. Now we swim in the sea that swept them away." [START_REF] Koolhaas | What Ever Happened to Urbanism[END_REF] Architects and urbanists are now like sailors set adrift without a helm. As Denise Scott Brown suggests, they'd best change their strategy: "The sailor may occasionally turn surfer, [and] ride the waves as they break." [START_REF] Venturi | Architecture as Signs and Systems: For a Mannerist Time[END_REF] How can they accept and participate in this suburban phenomenon that has spread out of control? In matters of city planning, this requires the admission that "the city no longer exists" and that the new reality requires abandoning the "twin fantasies of order and omnipotence" to embrace "staging of uncertainty". [START_REF] Koolhaas | [END_REF] Architects must let go of grand visions as it is too humiliating for them to set the goal of an ideal city only to witness its inevitable decomposition under all of the existing constraints.

When one has little or no control, it is better to see form as a destination rather than a starting point. In the suburbs of Utrecht in the Netherlands, the Maxwan firm had a similar experience: "Leidsche Rijn is an urbanism of negotiation, and proud of it. The negotiations were not done in order to get the design realised; the design was made to negotiate with, to get the city built. The most important ingredient of this story is that the urban designers ... had no real power at all. They did not even have a strong and stable power base to operate from. ... Having no power them-selves, their freedom of movement was not defined by the limits of their man-date." [START_REF]Orgware", Big Soft Orange[END_REF] For architecture, the same applies. Rather than fighting against all odds to impose refined construction that partners are unable or unwilling to build, it might be preferable to develop a specific architecture from the edge condition. Consider the example of limited budget: Instead of a stifling factor, it can be a source of creativity, as regularly shown by Anne Lacaton and Jean-Philippe Vassal. Similarly, mass-production of building components is not such a tragedy if one is inspired and makes them a central feature in the de-sign like Jacques Ferrier. And there are a thousand other ways to incorporate the edge condition into the very heart of the design process.

The nature of buildings and cities is dependent upon the social, technical and cultural context surrounding those who design them. In today's world, the architect's place is on the edge. That is, the architect no longer has a central place in society, and architecture is no longer the focal point in the urbanised field. The collectivity has other priorities. Still, our environment changes daily and architects and urbanists remain major players in this process. We can only hope that, from the edge, they will capture the spirit of their time as un-welcoming as it may be, creating their own unique and noteworthy interpretation.

V.D.


S Bougleux

O Lézoray

A Nouri

3D COLORED MESH STRUCTURE-PRESERVING FILTERING WITH ADAPTIVE P-LAPLACIAN ON DIRECTED GRAPHS

Keywords: 3D colored meshes, graph signal, structure preserving filtering, p-Laplacian

Editing of 3D colored meshes represents a fundamental component of nowadays computer vision and computer graphics applications. In this paper, we propose a framework based on the p-laplacian on directed graphs for structure-preserving filtering. This relies on a novel objective function composed of a fitting term, a smoothness term with a spatially-variant pTV norm, and a structure-preserving term. The last two terms can be related to formulations of the p-Laplacian on directed graphs. This enables to impose different forms of processing onto different graph areas for better smoothing quality.

INTRODUCTION

Structure-preserving filtering is one of the most important processing tasks dedicated to image editing and computational photography. During the last decade, many structurepreserving smoothing filters have been proposed so far [START_REF] Zhang | Rolling guidance filter[END_REF][START_REF] Cho | Bilateral texture filtering[END_REF][START_REF] Gastal | Domain transform for edge-aware image and video processing[END_REF][START_REF] Xu | Image smoothing via L 0 gradient minimization[END_REF]. To enable the editing of an image, they decompose it into a base layer containing the rough structures with preserved edges and several layers of increasing level of details. From this hierarchical representation, image editing tasks such as smoothing, abstraction, and sharpening can be performed [START_REF] Xu | Structure extraction from texture via relative total variation[END_REF]. At the same time, the advent of low cost 3D sensors has enabled the large development of 3D scanning. Using either a real 3D scanner or a set of images obtained by moving around an object, one can easily obtain 3D colored point clouds or meshes where each point or vertex is described by its 3D position and RGB color. However, the visual quality of the acquired data is not always of sufficient especially when a further 3D printing is planned. In addition, if the 3D data has to be used in virtual reality applications, it often has to be post-processed to be visually simplified. With such 3D processing tasks to be performed, there is interest into the development, as this has been done for images, of structure-preserving filters that operate on 3D colored data. Very few works have considered this kind of problem. Some structure-preserving filters have been extended to 3D meshes but they only consider the editing of the 3D points' coordinates to sharpen the mesh [START_REF] Fleishman | Bilateral mesh denoising[END_REF][START_REF] Kolomenkin | Prominent field for shape processing and analysis of archaeological artifacts[END_REF]. Some recent works have considered the extension for 3D colored meshes by a 3D extension of unsharp masking [START_REF] Afrose | Mesh color sharpening[END_REF] or with the use of morphological filters to obtain a hierarchical decomposition of the 3D color information [START_REF] Lézoray | Manifold-based mathematical morphology for graph signal editing of colored images and meshes[END_REF][START_REF] Lézoray | 3d colored mesh graph signals multi-layer morphological enhancement[END_REF] .

In this paper we propose a structure-preserving filter based on an objective function composed of three terms (Sec. 3): a data-fitting term, a smoothness term and a structure preserving term. To have a more adaptive filtering, the smoothness term relies on a spatially-variant p-total variation norm (pTV). This extends our previous works [START_REF] Abu-Aisheh | plaplacian regularization of signals on directed graphs[END_REF] that considered only the two first terms with a fixed pTV norm. The smoothness term can be related to formulations of the p-Laplacian on directed graphs (Sec. 2). Finally the structure-preserving term enables to preserve the most important structures. This enables to impose different forms of regularization onto different 3D areas for better smoothing quality, as illustrated and discussed in Sec. 4.

P-LAPLACIANS ON DIRECTED GRAPHS

Graphs provide a powerful and common structure for representing and processing structured and unstructured data, like images, colored 3D point clouds or meshes. The data are assumed to be composed of elements living in R nc , e.g. 3D points or RGB colors. A data of n v elements is represented by a real matrix F = [f i,c ] i∈Nv,c∈Nc , with N v = {1, . . . , n v } and N c = {1, . . . , n c }, so that f i,c encodes the c-th component of the i-th element. Alternatively, the i-th element can be represented by a vector f i = [f i,c ] c∈Nc . The elements are connected by edges to form a graph used as a domain for processing the data. Since connections do not usually have the same strength, a weight is associated to each edge, and the graph can be fully represented by its weighted vertex-vertex adjacency matrix W = [w i,j ] i∈Nv,j∈Nv , with w i,j ∈ (0, +∞) if the elements i and j are connected by an edge from i to j, or w i,j = 0 if they are not. Graphs are here assumed to be without self-loops (w i,i = 0) and directed (w i,j and w j,i can be different). This is motivated by the construction of graphs based on k-nearest neighbors, that are known to produce asymmetric connections in general, i.e. an edge from i to j does not necessarily imply an edge from j to i. Such graphs are commonly symmetrized, i.e. W is replaced by 1 2 (W + W T ), before processing the data. While this is mathematically correct for most methods based on the unnormalized Laplacian L = diag((W+W T )1 nv )-(W+W T ) [START_REF] Chung | Laplacians and the cheeger inequality for directed graphs[END_REF][START_REF] Hein | Graph laplacians and their convergence on random neighborhood graphs[END_REF][START_REF] Olfati-Saber | Consensus problems in networks of agents with switching topology and time-delays[END_REF], this is not the case for most second-order operators, especially non-linear operators like curvatures and p-Laplacians [START_REF] Zhou | Semisupervised learning on directed graphs[END_REF][START_REF] Abu-Aisheh | plaplacian regularization of signals on directed graphs[END_REF]. These operators are the main ingredients of filtering techniques based on variational principles, but few of them have been considered for directed graphs. We have recently proposed several formulations of the p-Laplacian [START_REF] Abu-Aisheh | plaplacian regularization of signals on directed graphs[END_REF]. In this paper we describe the unnormalized case used by the filtering process presented in the next section.

Given a data F, its gradient over an weighted graph W can be defined as

∇ W F = (∂ W j F) j∈Nv , with ∂ W j F = [ √ w i,j (f j,c -f i,c
)] i∈Nv,c∈Nc the directional differences according to vertex j. The gradient at a vertex i is thus given by

∇ W i F = [ √ w i,j (f j,c -f i,c )] j∈Nv,c∈Nc . Its magnitude |∇ W i F| =   nv j=1 w i,j nc c=1 (f j,c -f i,c ) 2   1/2 =   nv j=1 w i,j f j -f i 2   1/2 = nc c=1 |∇ W i F c | 2 1/2 , with f = nc c=1 f 2 c
and F c ∈ R nv the restriction of F to the c-th component, provides a basic tool for measuring the variations of F at each vertex. The regularity of F over the graph can then be measured by its p-total variation (pTV) defined as the L 1 norm of a power of the gradient magnitude:

F pTV = |∇ W F| p 1 = nv i=1 |∇ W i F| p (1)
where p ∈ [1, +∞) controls the degree of regularity, and

|∇ W F| = [ |∇ W i F| ] i∈Nv .
As for undirected graphs [START_REF] Elmoataz | Nonlocal discrete regularization on weighted graphs: A framework for image and manifold processing[END_REF][START_REF] Bougleux | Local and nonlocal discrete regularization on weighted graphs for image and mesh processing[END_REF], it is easy to show that ∇ F pTV = p L p,F F holds, with

L p,F = diag((W p,F + W T p,F )1 nv ) -(W p,F + W T p,F ) (2)
and W p,F = diag(|∇ W F| p-2 )W. In other terms, we have:

[L p,F F] i,c = nv j=1 w i,j |∇ W i F| 2-p + w j,i |∇ W j F| 2-p (f i,c -f j,c )
This defines the p-Laplacian of F on W. The Laplacian L is retrieved for p = 2, and for p = 1, L 1,F F defines the weighted (mean) curvature of F. The p-Laplacian is non-linear, and L p,F is symmetric, positive semi-definite, and can be viewed as a data-dependent Laplacian. It can be rewritten as L p,F = diag(W p,F 1 nv ) -W p,F with W p,F = W p,F + W T p,F , equivalent to a data-dependent Laplacian on an undirected graph.

The regularity measured by the pTV norm and the p-Laplacian depends on the global parameter p. As proposed in the following section, more flexibility can be obtained by adapting this parameter locally at each vertex.

ADAPTIVE P-LAPLACIAN STRUCTURE-PRESERVING FILTERING

We present a structure-preserving smoothing filter based on an adaptive p-Laplacian and an adaptive preservation of gradient magnitudes. This is inspired by a recent work in image processing [START_REF] Fan | Image smoothing via unsupervised learning[END_REF] which considers other measures of local variations and a different formulation of their local adaptation.

Energy formulation

Given a data F 0 ∈ R nv×nc , the proposed method consists to find a smoother version that minimizes an objective function

E(F, F 0 ) = λ d E d (F, F 0 )+λ s E s (F, F 0 )+λ r E r (F)
, with E d a data term, E r a regularity term, E s a structure-preserving term, and λ d , λ s , λ e ∈ [0, +∞) constant balancing weights.

The data term E d aims at providing a solution close to F 0 . This is measured according to the mean square error:

E d (F, F 0 ) = F -F 0 2 (3)
with F = tr(F T F). The local differences (f i,c -f 0 i,c ) should be null, or low, at vertices representing parts of the data that must be preserved. This is guided by the structurepreserving term E s . It measures a weighted mean square error between the gradient magnitude of F and the one of F 0 on the same weighted graph S = [s i,j ] i,j∈Nv :

E s (F, F 0 ) = 1 2n s nv i=1 α i |∇ S i F| 2 -|∇ S i F 0 | 2 2 (4) with α = [α i ] i∈Nv , α i ∈ [0, 1]
the degree of structure preservation, and n s = n c nv i=1 α i a normalization term. The term α behaves as a mask that enforces more or less the preservation of the gradient magnitude of F 0 . To ensure that structures in F 0 are preserved , it should be defined from an indicator or a detector of saliency, as detailed in Sec. 3.3.

The local differences involved in the approximation terms E d and E s should be higher at vertices representing unimportant details. This is obtained by reducing the variations of F 0 at these vertices according to an adaptive pTV function:

E r (F) = nv i=1 1 p i |∇ W i F| pi (5)
where p i ∈ [1, 2] fixes the degree of regularity at vertex i and ensures convexity of ( 5). Intuitively, it should depend on F 0 and be inversely proportional to α i , i.e. large at vertices where the data needs to be smoothed and low where the data must be preserved. Note that the graphs W and S are different.

In practice, it is more coherent and computationally effective that they are structurally equivalent, i.e. s i,j = 0 ⇔ w i,j = 0. Since the terms E d , E s and E r are convex functions of F, the objective function E is also convex, and several methods can be used to compute a solution. In this paper we derive a simple filter related to weighted mean filtering.

Filtering process

To optimize the objective function E w.r.t. F, we consider the gradient ∇E(F) = [∂E(F)/∂f i,c ] i∈Nv,c∈Nc and try to solve the system of nonlinear partial difference equations ∇E(F) = 0 with

∇E(F) = 2(F -F 0 ) + λ r L r,F F + 2
ns λ s L s,F F (6) where L r,F and L s,F are defined below.

Indeed, it is straightforward that ∇E d (F) = 2(F -F 0 ), and the gradient of the regularity term E r is given by ∇E r (F) = L r,F F, with

[Lr,FF]i,c = nv j=1 wi,j |∇ W i F| 2-p i + wj,i |∇ W j F| 2-p j (fi,c -fj,c)
Contrary to the p-Laplacian described in Sec. 2, L r,F uses here a regularity degree p = [p i ] i∈Nv adapted to each vertex. Similarly, the gradient of the structure preserving term E s is given by ∇E s (F) = 2 ns L s,F F with

[L s,F F] i,c = nv j=1 (a i s i,j + a j s j,i ) (f i,c -f j,c ) (7)
with

a i = α i (|∇ S i F| 2 -|∇ S i F 0 | 2 ).
We propose to use a linearized Gauss-Jacobi iterative method to find a solution to the system of non-linear equations. Let t be an iteration step, and F (t) be the solution at step t. Starting with F (0) = F 0 , and g 0 = |∇ S F 0 | 2 , the method iterates the following steps:

∀i, h i ← |∇ W i F (t) | pi-2 ∀i, g i ← α i |∇ S i F (t) | 2 -g 0 i ∀i, ∀j, w i,j ← 1 2 (h i w i,j + h j w j,i ) ∀i, ∀j, s i,j ← 1 ns (g i s i,j + g j s j,i ) ∀i, ∀c, f (t+1) i,c ← λ d f 0 i,c + nv j=1 (λ r w i,j + λ s s i,j )f (t) j,c
λ d + nv j=1 λ r w i,j + λ s s i,j until convergence or a given number of steps is reached. An iteration computes F (t+1) , at each vertex, as a weighted mean of F (t) in the neighborhood of the vertex in the graph W + S. Contrary to the family of filters based on pTV on undirected [START_REF] Elmoataz | Nonlocal discrete regularization on weighted graphs: A framework for image and manifold processing[END_REF][START_REF] Bougleux | Local and nonlocal discrete regularization on weighted graphs for image and mesh processing[END_REF] and directed graphs [START_REF] Abu-Aisheh | plaplacian regularization of signals on directed graphs[END_REF], the proposed filter adapts the regularity parameter p locally and enforces the gradients to be preserved according to a structure-preserving map.

Parameters for colored 3D point clouds

The proposed structure-preserving filter depends on several parameters: the graphs W and S, the vector of local regularity degrees p, and the vector of local structure-preserving strengths α. They are determined by the nature of the data and the desired filtering effect. We describe them for smoothing colored 3D point clouds. Given a set X = {x i } i=1,...,nv of n v points x i ∈ R 3 , and the RGB colors F 0 ∈ R nv×3 associated to these points (n c = 3), we assume a connected graph S 0 has already been constructed for connecting the points. When X samples the surface of a 3D object, S 0 should be the graph induced by the edges of a mesh having X as vertices. But any graph can be used as long as it is connected and represents the geometric structure of the data.

The graphs S and W are constructed by connecting each point x i ∈ X to its k nearest points, within an α-hop N α i in S 0 , according to the dissimilarity measure d(

x i , x j ) = d EMD (H(Φ τ i ), H(Φ τ j ))
, where Φ τ = (Φ τ i ) i=1,...,nv associates to each point x i a feature Φ τ i = (f 0 j ) j∈N τ i ∪{i} , i.e. the colors of the points x j around x i within a τ -hop N τ i in S 0 . Since S 0 is not assumed to be regular, features do not have the same size, and a simple L 2 norm cannot be used to compare them. The Earth Mover Distance d EMD [START_REF] Rubner | The earth mover's distance as a metric for image retrieval[END_REF] between the histograms H of the features in the L * a * b * color space is more appropriate. The retained connections are then directly used to construct the graph S. This graph is unweighted, i.e. s i,j = 1 if x j is among the k nearest neighbors of x i , or 0 otherwise. The graph W has the structure of S but it is weighted according to the distances between the features defined above by a parameter-less similarity:

w i,j = 1 - d EMD (H(Φ τ i ), H(Φ τ j )) max l=1,...,nv s i,l =1 d EMD (H(Φ τ i ), H(Φ τ l )) (8)
if s i,j = 1, and 0 else. Remark that S and W are directed.

The two other parameters p and α are defined from a common saliency map m = [m i ] i=1,...,nv which indicates, for each point x i , the local degree of structure in F 0 . This is measured by the normalized sum of distances within a ρ-hop in S 0 :

m i = 1 |N ρ i | j∈N ρ i d EMD (H(Φ ρ-1 i ), H(Φ ρ-1 j
)). The parameters are then defined by :

p i = 1 + 1 1 + m 2 i , α i = m i -min j m j δ i (max j m j -min j m j ) (9)
with δ i the number of edges starting at i in S (outgoing degree). This enables to normalize the gradients in the objective function E s . As discussed in the previous section, p and α defined by Eq. 9 are antagonists, one for smoothing the data and the other for preserving its main structures.

EXPERIMENTAL RESULTS AND CONCLUSION

To illustrate the proposed feature-preserving filter, we consider colored 3D point clouds acquired by digitizing an object by means of a scanner. A mesh obtained by reconstruction techniques is also provided. We use this mesh to define the initial graph S 0 . For all the experiments, the parameters for constructing the graphs S and W, and the saliency map m, are fixed to α = 5, τ = 1 and ρ = 2 for the sizes of the hops, and the number of neighbors is set to k = 10. The balancing terms in the objective function are set to λ d = 10 -3 , λ r = 1, and λ s = 0.25. The number of iterations is fixed to 25.

Figure 1 shows results on a low-resolution mesh (a scan of a stuffed duck with 19247 vertices and 38490 faces) to illustrate the benefit of the proposed approach (each image shows a caption of the mesh and two cropped and zoomed areas). First row shows the original mesh and its structure mask M (shown with a heat map LUT). Second row shows classical results obtained with fixed values of p and no structurepreserving term. With p = 2 (the Laplacian) a strong smoothing effect is obtained, whereas with p = 1 (the Graph To-Fig. 2. First row: original mesh, structure mask. Second row: filtering and sharpening results. tal Variation), the edges are better preserved. However some small structures are also kept. Better filtering results are obtained with a spatially varying norm and the unwanted small details are removed. However, some parts such as the duck's collar are now more blurry. This problem is then corrected by using the whole model we propose that incorporates an additional structure-preserving term. The final result is inline with our objective: to eliminate unimportant fine-scale details while maintaining the primary structures.

Figure 2 shows an exemple of 3D colored mesh editing for sharpening. The mesh is a high-resolution scan (780977 vertices and 1557701 faces) of a medieval house in the city of Bellac, France. As it can be seen, in the filtering result, the small fine details have been suppressed while preserving the strongest structures. This can be used as the base layer of a sharpness enhancement procedure [START_REF] Farbman | Edge-preserving decompositions for multi-scale tone and detail manipulation[END_REF]. The difference between the original mesh and its filtering is computed, boosted by a factor 0.8 and added back to the filtered version. As it can be seen, the sharpened result has enhanced the local contrast without artifact magnification or detail loss. Note that the enhancements are better visible at high resolution. This shows the benefit of our approach for 3D editing tasks. Further works will consider other minimization schemes as well as other editing tasks such as inpainting and abstraction.

Fig. 1 .

 1 Fig. 1. Illustration of the proposed model.

This work received funding from the Agence Nationale de la Recherche (ANR-14-CE27-0001 GRAPHSIP), from the European Union FEDER/FSE 2014/2020 (GRAPHSIP project), and the support of NVIDIA Corporation with the donation of a Titan Xp GPU used for this research.