Hélène Fleury

From Marx to Counterculture. The Marxian Vision of Art(ist) and Véquaud's Maithil Village Communitarian Utopia: A Renewed Romanticism?

The Marxian vision of art differs greatly from socialist realism and even from the neo-(if not post-)Marxist philosophy of critical theorists like Adorno and the Frankfurt School. Counter to the Biedermeier uprightness, Marx will give to the art a real social content. The artist's work is elevated to an ideal labor organization and opposed to the worker's alienation resulting from mechanization and the division of labor. If a society based on the model of artistic work remains a utopia, Marx seems to be seeking to renew the ancient model of the medieval craftsman. The latter is free, masters his trade, skills and techniques, and succeeds in his work as an artists. If this notion is very different from the idea based on the socialist realism, it meets, on the other hand, characteristics of the countercultureas the refusal of the work as socially imposed -even if "Flower Children" and other agents of the counter-culture rarely recognize themselves in a Marxian vision of art. It is the case of Yves Véquaud, following on the heels of the counter-culture, while not assuming any clear political commitment. Inspired by a counter-bourgeois hippie Bohemia, he extols in his writings some heroïzed figure of painters, with neo-tantric elements, embodied in the bucolic communitarian Maithil village utopia, through which the Sehnsucht of lost paradise occurs. Would Véquaud, within his artistic vision, be a Marxian without being aware of it? However, his depiction of the bucolic Maithil village and its myriad of women-artists seems (incidentally?) be inspired by the Marxian utopia of a society based on the model of artistic work. The common feature between the counter-culture and the Marxian notion of art lies most certainly in the 'revolutionary and/or utopian romanticism' defined by Löwry & Sayre.

Véquaud was a French writer, a contributor to the Nouvelle Revue Française, a translator, a film director, and a curator, today mostly known for his essays on Mithila paintings. In the present communication we aim to show the indirect impact of Marx's ideas on Véquaud as a player of the countercultural indophilia. We hypothesize that in the case of Véquaud the traces of Marx go beyond the ideological references of the counter culture. We will try to assert this hypothesis in particular with the similarity of vision between both authors concerning the so-called "communitarian Indian village", as a counter-model challenging capitalist society, but stemming from a persistent Western orientalist imaginary and primitivist schema.

Marx's aesthetics: an overview

Marx as a romantic poet (1835-37)

As a poet, Marx was influenced in particular by E.T.A. Hoffmann's and Fichte's romanticism and his academic studies about ancient literature and German classical aesthetics. The young Marx wrote poems inspired by the love for his fiancée, but his poetry treats also romanticist themes as the primacy of feeling and human emotions, the situation of artists separated from "normal" life, the struggle against philistines, or the opposition to an abstract and dehumanized world.

Marx as early philosopher: organic unity in the material and social worlds (1837-41)

In 1837, Marx left his vocation of poet to become philosopher. He combines poetry and prose, art and knowledge, fields that he considered to have been separated for a too long time 1 . So, it is not surprising that he read Hegel: as a representative of Naturphilosophie, the latter aims to reunifie everything, but under the metaphysics of the Absolute. Marx was against the abstraction of Hegel's philosophy that he wished to escape.

The then widespread notion of organic unity is playing a crucial role in his oeuvre. In his dissertation on late Greek philosophy 2 , the atom is regarded as the material basis of the world, but it symbolizes also individual self-consciousness. Within a complex body, the atoms cannot be seen isolated from one another. As a higher unit, this complex body bears witness to the diversity of the world and its societies. In this perspective, Marx disapproved the abstract principles of the French Constitution of 1783 that may lead to a 'world of uniform and independent atom-citizens' [START_REF] Lifshitz | The Philosophy of Art of Karl Marx[END_REF] , far away from the living forces of society as essential entity for human beings. 1.3. Marx's 'not written aesthetics': the artist create freely organic forms in an organic society (1841/42)

After a life-path that leads him from poetry to philosophy, Marx would have been well placed to write an aesthetics, what he did. With Bruno Bauer he worked in 1841/2 on a critique of Hegel's vision of art and religion. But the two works written within this continuity in 1842 on Christian art are not extant [START_REF]Über christliche Kunst & Über Religion und Kunst mit besonderer Beziehung auf christliche Kunst[END_REF] . So it is understandable that Marxian aesthetics can be described as 'a not written aesthetics' [START_REF] Hans | Kritik der marxistischen Kunsttheorie[END_REF] . However, art is evoked here and there in Marx's oeuvre, and his approach to art is often integrated into more general considerations. The notebooks of 1842 can give some few ideas about the supposed content of his not extant works on Christian art. Those notebooks introduce the idea of fetishism applied to religion: the latter worships the materiality of things. In contrast, art seeks to create organic forms through the intermediary of imagination. For Marx, this is how to explain the dryness of religious art [START_REF] Lifshitz | The Philosophy of Art of Karl Marx[END_REF] . The origin of organic conception of art lies in a free society which have to be organic too.

It is interesting that Marx distinguished two worlds governed by the same organic principal, freely coexisting one in relation which each other: the real and the artistic worlds. Is there not a similarity with the early romanticist "two-worlds" model (zwei-Welten Modell) [START_REF] Cf | Musik im Abendland: Prozesse und Stationen vom Mittelalter bis zur Gegenwart[END_REF] ? In that model, both universes are not parallel, but interconnected: the negation of the real world provides access to the one of arts and music, which in turn retroacts onto the real world. The main difference between the early romanticist vision and the Marxian one is the fact that the latter did not oppose a poetic world, seen in positive terms, to a trivial one, judged negatively. Marx is more and more attached to the reality perceived through our senses and give less and less importance to idealism.

The first political economical works : alienation as emerging concept (1844/45)

In 1844/45, Marx wrote his first political economical works: the Parisian manuscripts and the Holy Family (the latter together with Engels). The Parisian manuscripts developed the notion of "alienated" and "estranged" work (entäußerte, entfremdete Arbeit), which is constitute by: 1. The estrangement of things: as working is not belonging to the worker's intrinsic nature, the production exercises power over him as an alien object; 2. Self-estrangement: the relation of the worker to his activity considered as not belonging to him, as an activity turned against him; 3. Man's speciesbeing: the worker as a being alien to him, to his own body, his external nature and human aspect; 4. The estrangement of man from man as a consequence of the estrangement from his production, his life activity and his species-being. Marx deplores that narrow-minded relationships leading to equally narrow-minded senses are left, after human being had improve his mastery of the five sense during a large part of the history of mankind.

From the German Ideology : towards a re-esthetized communist society (1845-)

Althusser regarded the German Ideology. The Theses on Feuerbach as an "epistemological break" (coupure épistémologique) between the "ideological" period of the Early Works and the "scientific" period after 1845. Furthermore, the German Ideology was canonized in the 1920s/30s as the founding text of the materialist conception of history. Surely, these periodisation and canonization can be easily deconstructed. Nevertheless, from 1845, Marx and Engels would criticize the aesthetic visions of the Young Hegeliansincluding Feuerbachand show the interdependence between, on one hand, the art and the creative spirit of the artist, on the other hand, the history of economic and politic life in society. They develop less an aesthetics than an analysis and critique of capitalism as a social system. For both authors, art and creativity are called upon to play a major role for the perspective of a new re-esthetized communist society in a time of disalienation.

In the ideal society, everybody is doing from day to day various activities, including arts. There is no more division of labour, even between artists and no-artists: 'In a communist society there are no painters but only people who engage in painting among other activities' [START_REF]fällt jedenfalls fort die Subsumtion des Künstlers unter die lokale und nationale Borniertheit, die rein aus der Teilung der Arbeit hervorgeht, und die Subsumtion des Individuums unter diese bestimmte Kunst, so daß es ausschließlich Maler, Bildhauer usw. ist und schon der Name die Borniertheit seiner geschäftlichen Entwicklung und seine Abhängigkeit von der Teilung der Arbeit hinlänglich ausdrückt[END_REF] . As long as this utopia would not be achieved, the economic structure of society may be not isolated from artistic production: hence the need for a mediation between base and superstructure.

Marx and Engels gave also examples of pre-capitalist societies, among others the ancient Greek society, the guilds organizations in Middle Ages, and the village community in India. The most significant for our purpose is Marx's vision, influenced by Hegel, of the "idyllic" Indian village community as a pre-capitalist immutable form of organisation. In the Capital, he stresses the prevalence of 'possession in common of the land' and the mode of production able to sustain a "natural economy" through the union of handicraft and agriculture. Only the surplus becomes a commodity when it circulates outside the village. Marx and Engels were looking for societies which are distant in time (Antiquity, Middle Ages) and space (India and other Asian countries), with a precapitalist mode of production. But for them, these essentialized visions are neither the reflection of an Golden Age nostalgia nor the dream of an elsewhere. They consider, in different steps, the way to break away capitalism, pursuing the ideal vision of communist society.

Vequaud's aesthetics and its similarities with Marx's ideas

Véquaud's early writings: traces of counter-cultural romantic revolt

From his first writings, Véquaud followed on the heels of antibourgeois Romanticism and critical thought:

-The extolling of emotions and body: the hippie movement called into question the self-control required to maintain one's rank or place, such as the body hexis. Emotions are liberated from the social constraints of daily-life lived as alienation. For example, in Le voyage en écriture, a forced hairdressing session is felt as a traumatic, symbolic amputation. An anti-conformist body hexis based on natural body postures is extolled. Later, his travel books on India are characterized by a "creative bubbling" [START_REF] Pagis | un pavé dans leur histoire[END_REF] stemming from a cult of pleasure and a hedonistic way of life. Exalted emotions, linked to experiences of transgression, and the questioning of social points of reference leads to a feeling that anything is possible. The feeling of festive (party) liberation and freedom are associated with a spiritual register. In 1974, the initiation given by the Dalai-lama was described as 'one of the most extraordinary party which you could see in the life' [START_REF] Véquaud | Lettre de Bodh Gaya: quand le dalaï lama donne une initiation[END_REF] . These literary fantasies entail colourful sensorial descriptions, allowing to open the 'field of the possibilities' (champ des possibles). The prevalence of affects matches with romantic hippie values, in opposition with cold and abstract knowledge, identified as stemming from colonization as antidote to romance and imagination.

-The critique of the values of bourgeoisie: Véquaud refused the social finiteness and the bourgeois mandate of remunerated work and family. In Le Voyage en écriture, life is represented as a train from which we must jump in order to escape from despondency 11 , reminding Timothy Leary. Véquaud questioned also the function of education, which socialized children to accept hierarchy, in particular in a provocative article entitled: 'The parents doesn't love their children, the proof is that they put them in school'. This rejection of educational connections leads him to give up teaching and to invest in the writing that he considered as a vocation, after being awarded in 1967 by the Félix Fénéon literary price. Thus, Véquaud experienced a broke and happy hippie bohemia.

-The praise of bohemia: If Véquaud was able to accumulate cultural and artistic capital, his economic one remained very fragile. But he defied the petty bourgeois way of life, from which he originated, to seize the liberality of manners, presumed associated with aristocracy and the upper-class, while experiencing precariousness. So, Véquaud's semi-autobiographical writings revived a myth of bohemia, inseparable from a vocational art definition, whose singularity regime reinforces the figure of the marginal genius. He evolved in a Bourdieusian "economic world turned upside down", depending on different modalities:  Regarding the means for subsistence: he managed situations which he presumed to be temporary, without too much concern for saving money;  Concerning the autonomy of the field of artistic creation, the narrator of his novels aspire to a freedom auspicious to creation. In Le Voyage en écriture, he scoffs at a dreary routine and enjoins to drop out: "Open your head, wash well your memory, . . . this story is not yours. Go through, go through" 12 .  Finally, as for posterity postponed, Yves Véquaud strove for glory and not for gain, contrary to the bourgeois artist who expects success, in particular financial one, and immediate notoriety.

Marx and Véquaud wished at first to make writing their vocation. Both appropriated common themes of Early Romanticism which are also those of counterculture: the exaltation of emotions, the critique of philistines or bourgeoisie, the praise of bohemia or the cursed artist at the margins of the society. The common denominator between both authors can be found in the notion of romanticism in the broader meaning of the sense forged by Löwy and Sayre [START_REF] Löwy | Le romantisme à contre-courant de la modernité[END_REF] . They define romanticism not only as a literary trend but more widely as an anti-establishment Weltanschauung against rationalization and commodification of human relationships.

Véquaud's communitarian Maithil village utopia

Since 1969, Véquaud spent two years in Nepal, India, and Ceylon, following the hippie trail. He discovered Mithila paintings in New-Delhi and went to Mithila several times to meet village painters. Véquaud expressed a canonical form of the counter-culture: the back-to-the-landmovement in communitarian utopias within the Maithil village. His fantasy model shares the suspension of dominant norms like phalansterie, monachisms, or communes involving: widened household, collective care of children, egalitarian rotation of domestic tasks. Gender, patriarchy, and family are redefined. He notes that men often do nothing else than keeping their lastborns on the knees [START_REF] Véquaud | En Occident, on peut compter sur les doigts d'une main les femmes peintres qui brillèrent par leur art. Ici, dans les deux hameaux où nous avons vécu, deux cents, trois cents femmes exécutent des merveilles[END_REF] . In West, 'one can count on the fingers of one hand the painting women who shine through their art. Here, . . . two hundred, three hundred women make marvels' 15 . According to him, in India, 'the example is not rare of societies where women are freer and more powerful than in ours' 16 . These projective presumptions, putting forward a phantasmagorical experience of gender resocialization, are reiterated in his writings. Here Mithila paintings contribute to "undoing gender" 17 , emphasizing the impact of feminist cause. This village utopia serves his quest of alternative prophecies.

It renews an epitomized unchanging tradition opposed to mass and technocratic society. The rise of "information age" is accompanied by nostalgia for older ways of making things. As the romantic critics of Victorian age praised Indian craftsmanship, Véquaud attributes the perfection of Maithil art to a timeless village. Folk art is misunderstood as homogenous, stable, and rural. He champions a return to a pre-industrial, holistic community. He calls to leave towns as Madhubani to find hamlets where the Maithil civilization stays alive. From his standpoint, the preserved culture of Mithila embodies the culture of India. His utopia expresses a romantic nostalgia for the Heimat embodied in an elsewhere, through which arise the Sehnsucht of a lost Eden. He attempts to reenchant the world. So does he by heroizing the women painters.

Yves Véquaud emphasizes in his writings a revealing interpretation of the icons of his utopian counter-society. The Maithil women painters are presented as prophetic figures of legend, between holiness and genius. From his standpoint, 'to paint better than her neighbour is to be devout, as it were to become priestess'. The artistic success of painters involves both the personalization of means as well as the depersonalization of the ends of the success, namely the creation of objects crystallizing values recognized beyond the author. It was traditionally reserved for heroes and saints. This literary fantasy bears witness to a picturesque vision of India within an orientalising episteme. Véquaud praises a sort of Maithil Republic of Arts, through a pastoral myth and art religion, prevalent during the first Romanticism. His narratives reflect a male Western style for representing the painters. They bore less relation to Maithil painters than to a crisis (in Western) self-representation.

There are many similarities between Marx's and Véquaud's visions of the Indian village: the idyllic and timeless character, the pre-industrial or pre-capitalist holistic community based on the union of arts / handicrafts and agriculture. If counterculture tends to free individuals from collectivity, the holistic model of the Indian village marks a return to Marx's notion of organic unity. Besides, indophilia prevails at Marx's and Véquaud's time. Marx was living during Raymond Schwab's "Oriental Renaissance". Reading the first published translations of Sanskrit, artists and intellectuals thought then to discover an echo of humanity's origins, nearest of an harmonious nature and the divine. The indophilia of the sixties underlines a similar cultural movement of romantic contestation, associated to a second "oriental renaissance", as underlined by Raphaël Rousseleau. At last, these two visions of the Indian community matches with Löwy's and Sayre's category of "revolutionary and/or utopian romanticism". The difference between both is that Véquaud's literary fantasy reveals more the nostalgia of an enchanted elsewhere, while Marx renewed precapitalist modelsas the Indian villagein order to lay the groundwork for the communist society of the future.

For the young Marx and Véquaud, freedom and especially artistic freedom is crucial. In this sense, socialist realism may be thought as the negation of the aesthetics of the Early Marx. On the other hand, countercultural orientalism as represented by Véquaud can be seen on a more obvious way as heterodox Marxisms as the negation of socialist realism. So we are tempted to say that Véquaud's vision is the "negation of the negation" of the ideas of the Early Marx. One more reason to explain the similarity of both, following respectively a Zeitgeist characterized by romanticism and indophilia.

Letter to his father, 10 November 1837.

Differenz der demokritischen und epikureischen Naturphilosophie nebst einem Anhange, Diss. Universität Jena, 1841.


Ginette Vagenheim

Pirro Ligorio

Apollo Killing

Jeffrey E Horvitz

Pirro Ligorio's Early Drawings and the Influence of Polidoro da Caravaggio: A Focus on Landscapes

During my stay at CASVA, I explored the validity and limits of John Gere's assumption, advanced in the journal Master Drawings in 1971, that the early draftsmanship of Pirro Ligorio (c. 1513 /1514 -1583) must be construed as completely dependent on the example of Polidoro da Caravaggio (c. 1499 -1543). Gere's ideas have been extremely influential, to the point where the subsequent identification and explanation of Ligorio's initial works as a draftsman have been based on his theoretical premises.

After a careful rereading of Gere's work, however, I realized that his hypotheses substantially rely on information provided in Giovanni Baglione's life of Ligorio, written in the first half of the seventeenth century. According to Baglione, during his early career in Rome (1534), Ligorio mostly decorated Roman palaces and houses with "trophies, friezes and stories illustrating the magnificenze romane" in the manner of Polidoro. Although all of Ligorio's facade decorations have disappeared, Gere came to the conclusion that Baglione's testimony was evidence enough to assume that Ligorio's designs were executed in conformity with the compositional patterns created by Polidoro in the 1520s. To be sure, Gere was able to correctly attribute to Ligorio some drawings traditionally credited to Polidoro, such as Victory with Roman Trophies (Kupferstichkabinett, Berlin) and Roman Trophies with Prisoners (Pierpont Morgan Library, New York), thereby enabling us to gain an idea of the appearance of the lost facade decorations and to assess the extent of Polidoro's influence on Ligorio's drawings. Moreover, Gere's essay offers a very precise description of the principal characteristics of Ligorio's draftsmanship, for instance, his quirky way of drawing the hands of his figures. As further proof of the link between Polidoro and Ligorio, Gere cited the imprint of Polidoro in the attitudes and dispositions of the figures represented in Dance of Salome (Oratorio di San Giovanni Decollato, Rome), the only extant fresco by Ligorio. Following this convincing analysis, Gere proceeded to search for and find Ligorio's early style in a group of drawings that he considered stylistically close to Polidoro but not by Polidoro himself, ascribing them to Ligorio on the basis of a general stylistic similarity or even the subject matter represented.

Philip Galle after Giulio Romano, published by Hieronymus Cock, Apollo and Artemis Killing Niobe's Children, 1557, engraving. The British Museum; © Trustees of the British Museum Even now, Gere's method of identification of the Ligorio drawings is systematically applied, and sometimes with manifestly erroneous results. This is the case for Diana and Apollo Killing the Children of Niobe, now in the National Gallery of Art, Washington. This work was attributed to Ligorio because it represents an episode treated in Polidoro's decoration on the facade of the Palazzo Milesi in Rome, but the style of the drawing is completely different from Ligorio's. In addition, the inventor of the composition was not Polidoro but Giulio Romano; in fact, the drawing is reproduced in reverse in an engraving by Philip Galle published in 1557 by Hieronymus Cock with the inscription IVLIVS MANTVA INVE. H. COCK. EXCVDE. 1557. PHILIPE GALLE FECIT. We also find mention of the copper plate in the inventory drawn up upon Cock's death on March 1, 1601: "een coperen plaete van de Schieters van Julius Mantuanus."

After pondering the assumptions on which Gere's attributions were based, I decided to systematically examine the catalog of Ligorio's early drawings in the Master Drawings article. It was my intention to gauge their authenticity on broader grounds, checking all the aspects of Ligorio's particular style. I have thus begun to define and circumscribe all the characteristics of his early style. To that end, I have focused on a group of works that is surely autograph; that is, the sixteen well-known drawings illustrating the story of Hippolytus preserved in the Pierpont Morgan Library, New York, which I was able to study during a trip to New York. The material gathered during my stay at CASVA will be employed in the first chapter of my monograph on Ligorio's drawings. "The Genesis of Ligorio's Style," which will serve as a general introduction to the volume.

Research carried out in the National Gallery of Art Library and image collections and the department of prints and drawings has radically modified my approach to Ligorio's draftsmanship. During my residency, I also had the unique opportunity to become acquainted with the three hundred drawings registered in the checklist drawn up by David Coffin, the greatest specialist on Ligorio, and published as an appendix to his posthumous Pirro Ligorio: The Renaissance Artist, Architect, and Antiquarian (2004) with the purpose of serving as the groundwork for a catalogue raisonné of Ligorio's drawings.


Julie Lapébie
email: julie.lapebie@univ-amu.fr

DEGREE FORMULAS FOR THE EULER CHARACTERISTIC OF SOME SEMIALGEBRAIC SETS

Keywords: 2010 Mathematics Subject Classication. 14P10, 14P25, 58K05 Morse theory, manifold with corners, Euler characteristic, semialgebraic sets

We are interested in computing alternate sums of Euler characteristics of some particular semialgebraic sets, intersections of an algebraic one, smooth or with nitely many singularities, with sets given by just one polynomial inequality. We state theorems relating these alternate sums of characteristics to some topological degrees at innity of polynomial mappings.

Introduction

Let F = (F 1 , . . . , F k ) : R n → R k and G = (G 1 , . . . , G l ) : R n → R l be polynomial maps with k, l ≥ 1 and let W be F -1 (0). We are interested in the semialgebraic sets

W G (ε) := W ∩ {(-1) ε 1 G 1 ≥ 0} ∩ • • • ∩ {(-1) ε l G l ≥ 0},
where ε = (ε 1 , . . . , ε l ) ∈ {0, 1} l , and we want to express the alternate sum ε∈{0,1} l (-1) |ε| χ(W G (ε))

in terms of the topological degree at innity of polynomial mappings involving F and G, with |ε| = l i=1 ε i . These semialgebraic sets are natural generalizations of those studied by Szafraniec in [START_REF] Szafraniec | On the Euler characteristic of analytic and algebraic sets[END_REF] and [START_REF] Szafraniec | The Euler characteristic of algebraic complete intersections[END_REF]. They are simply dened by equalities and inequalities. If we want to use Morse theory for manifold with corners, they are the natural objects to study. Moreover when W = R n , these sets are basics semialgebraic sets. To give an idea, when l = 1, this alternate sum is equal to

(-1) |0| χ W ∩ {(-1) 0 G 1 ≥ 0} + (-1) |1| χ(W ∩ {(-1) 1 G 1 ≥ 0}) = χ W ∩ {G 1 ≥ 0} -χ W ∩ {G 1 ≤ 0} ;
and when l = 2, it is equal to

(-1) |(0,0)| χ W ∩ {(-1) 0 G 1 ≥ 0} ∩ {(-1) 0 G 2 ≥ 0} +(-1) |(1,0)| χ(W ∩ {(-1) 1 G 1 ≥ 0} ∩ {(-1) 0 G 2 ≥ 0}) +(-1) |(1,1)| χ(W ∩ {(-1) 1 G 1 ≥ 0} ∩ {(-1) 1 G 2 ≥ 0}) +(-1) |(0,1)| χ W ∩ {(-1) 0 G 1 ≥ 0} ∩ {(-1) 1 G 2 ≥ 0} = χ W ∩ {G 1 ≥ 0} ∩ {G 2 ≥ 0} -χ(W ∩ {G 1 ≤ 0} ∩ {G 2 ≥ 0}) +χ(W ∩ {G 1 ≤ 0} ∩ {G 2 ≤ 0}) -χ W ∩ {G 1 ≥ 0} ∩ {G 2 ≤ 0} .
There are already several results relating the Euler characteristic of an algebraic set to the topological degree of a polynomial mapping. First, when W is compact, Bruce [START_REF] Bruce | Euler characteristics of real varieties[END_REF] and Szafraniec [START_REF] Szafraniec | On the Euler characteristic of analytic and algebraic sets[END_REF] proved that there exists a polynomial P : R n+1 → R with an isolated critical point at the origin such that

χ(W ) = 1 2 (-1) n -deg 0 ∇P ,
where deg 0 ∇P is the topological degree at the origin of the gradient of P . In all this paper, we will denote by deg ∞ φ the topological degree at innity of φ φ :

S N R → S N 1 for any mapping φ : R N +1 → R N +1 such that φ -1 (0) B N +1 R . When k < n and W is a smooth (n -k)-dimensional manifold, Szafraniec constructs in [6] a polynomial map H : R n+k → R n+k such that H -1 (0) is compact and such that χ(W ) = (-1) k deg ∞ H.
In [START_REF] Dutertre | On topological invariants associated with a polynomial with isolated critical points[END_REF], Dutertre studies semialgebraic sets with isolated singularities. From f : R n → R a polynomial function with isolated critical points such that f (0) > 0, he constructs four polynomial mappings H, L, L 1 and L 2 in terms of f and ∇f such that H -1 (0), L -1 (0), L -1 1 (0) and L -1 2 (0) are compact. Then he gets the equalities:

• if n is even, then χ(f -1 (0)) = deg ∞ H + deg ∞ ∇f -deg ∞ L 2 χ({f ≥ 0}) -χ({f ≤ 0}) = 1 -deg ∞ L + deg ∞ L 1 . • If n is odd, then χ(f -1 (0)) = deg ∞ L -deg ∞ L 1 χ({f ≥ 0}) -χ({f ≤ 0}) = 1 -deg ∞ H -deg ∞ ∇f + deg ∞ L 2 .
The aim of this paper is to obtain a similar result as Szafraniec's one in [START_REF] Szafraniec | The Euler characteristic of algebraic complete intersections[END_REF] for our sum l i=1 (-1) |ε| χ(W G (ε)). Then we generalize the results of Dutertre in [START_REF] Dutertre | On topological invariants associated with a polynomial with isolated critical points[END_REF] to the sets

W G (ε) ∩ {f ≥ 0}, W G (ε) ∩ {f ≤ 0} and W G (ε) ∩ {f = 0}, when W = R n , namely we compute the alternate sums ε∈{0,1} l (-1) |ε| χ(W G (ε) ∩ {f = 0}) and ε∈{0,1} l (-1) |ε| χ(W G (ε) ∩ {f ≥ 0}) -χ(W G (ε) ∩ {f ≤ 0}) .
We suppose W = F -1 (0) smooth with F = (F 1 , . . . , F k ), G = (G 1 , . . . , G l ) and k + l ≤ n, and we assume that all the intersections between the set W = F -1 (0) and the sets G -1 i (0), i = 1, . . . , l are transverse (see Condition ( ) on Page 3). We will see that we can write ε (-1) |ε| χ(W G (ε)) in terms of the topological degree at innity of the following mapping L dened on R n × R k × R l by:

L(x, λ, µ) = ∇ω(x) + k i=1 λ i ∇F i (x) + l j=1 µ j ∇G j (x), F (x), µ 1 G 1 (x), . . . , µ l G l (x) , where ω(x) = 1 2 (x 2 1 + • • • + x 2 n ).
More precisely, we nd the result (c.f. Theorem 2.7):

ε∈{0,1} l (-1) |ε| χ(W G (ε)) = (-1) k deg ∞ L.
To prove this theorem, we will use Morse theory for manifolds with corner, which is an application of stratied Morse theory ([3], [START_REF] Hamm | On stratied Morse theory[END_REF]). For more details on this theory, we refer to [START_REF] Dutertre | On topological invariants associated with a polynomial with isolated critical points[END_REF].

In my thesis, I also obtained similar results for semialgebraic sets with isolated singularities.

The paper is organized as follows: in Section 2, we prove some technical lemmas that relate a Morse index to a topological degree, in order to give a similar result as the one of Szafraniec in [START_REF] Szafraniec | The Euler characteristic of algebraic complete intersections[END_REF].Some computations are given in the example at the end of each section. They have been done with a program written by Andrzej Lecki. The author is very grateful to him and Zbigniew Szafraniec for giving her this program.

Smooth case

In this section, we compute the alternate sums :

ε∈{0,1} l (-1) |ε| χ(W G (ε)),
and as Szafraniece in [START_REF] Szafraniec | The Euler characteristic of algebraic complete intersections[END_REF], we want to write it in terms of the topological degree of a polynomial mapping.

Let F = (F 1 , . . . , F k ) : R n → R k and G = (G 1 , . . . , G l ) : R n → R l be polynomial maps, with k, l ≥ 1 and k + l ≤ n. We dene W = {x ∈ R n | F (x) = 0}
and assume it is a smooth manifold of dimension n -k in R n (or empty). We also make the following transversality assumption:

( ) ∀{i 1 , . . . , i s } ⊆ {1, . . . , l}, ∀x ∈ W ∩ G -1 i 1 (0) ∩ • • • ∩ G -1 is (0), rank (DF (x), DG i 1 (x), . . . , DG is (x)) = k + s.
We are interested in the topology of the set

W G (ε) := W ∩ {x ∈ R n |(-1) ε 1 G 1 (x) ≥ 0} ∩ • • • ∩ {x ∈ R n |(-1) ε l G l (x) ≥ 0}
for ε ∈ {0, 1} l and in computing the value of the alternate sum

ε∈{0,1} l (-1) |ε| χ(W G (ε)), where |ε| = l i=1 ε i .
Let us remark that for ε ∈ {0, 1} l , W G (ε) is a manifold with corners, so it is also a Whitney stratied set, whose 2 l strata are given by:

S ε I j = x ∈ W | (-1) ε i G i (x) > 0 if i ∈ I j and G i (x) = 0 if i ∈ I j
where for all j ∈ {1, . . . , l}, I j is a subset {i 1 , . . . , i j } of {1, . . . , l} such that i 1 < • • • < i j , and where we write I j its complement in {1, . . . , l}. Moreover, we denote by I 0 = ∅, and then we have I 0 = {1, . . . , l}. For example, when l = 2, G = (G 1 , G 2 ) : R n → R 2 and for a given ε = (ε 1 , ε 2 ) ∈ {0, 1} 2 , the four strata are:

S ε I 0 = {x ∈ W | G 1 (x) = G 2 (x) = 0} , S ε I 1 = {x ∈ W | (-1) ε 1 G 1 (x) > 0, G 2 (x) = 0} , S ε I 1 = {x ∈ W | (-1) ε 2 G 2 (x) > 0, G 1 (x) = 0} , S ε I 2 = {x ∈ W | (-1) ε 1 G 1 (x) > 0, (-1) ε 2 G 2 (x) > 0} .
We will also consider the following submanifold of W :

W I j = x ∈ W | G i (x) = 0 if i ∈ I j , G i (x) = 0 if i ∈ I j
, and so we have that for all ε ∈ {0, 1} l and for all j ∈ {0, . . . , l}, S ε I j ⊆ W I j . We denote by W I j the closure of W I j in W , that is

W I j = x ∈ W | G i (x) = 0 ∀i ∈ I j .
Given a polynomial function ρ from R n to R, we are rst going to study the critical points of ρ on the dierent strata. We need for this to recall the denition of a correct critical point on a manifold with corners. Denition 2.1. Let M ∩ {h 1 ≥ 0, . . . , h k ≥ 0} be a manifold with corners and let us set

H := M ∩ {h i 1 = 0, . . . , h is = 0} ∩ {h i s+1 > 0, . . . , h i k > 0}. We say that a critical point p of ρ |H is correct if p is not a critical point of ρ on M ∩ ∪ i∈I {h i = 0}
for any proper subset I of {i 1 , . . . , i s }. Moreover, we say that p is pointing inward (resp. outward) the set

M ∩ {h 1 ≥ 0, . . . , h k ≥ 0} if α i 1 > 0, . . . , α is > 0 where ∇ρ(p) = s j=1 α i j ∇h i j (p) (resp. α i 1 < 0, . . . , α is < 0).
We remark that in our case, p ∈ S ε I j is a correct critical point of ρ |W G (ε) if and only if p is a critical point of ρ| W I j and is not a critical point of ρ |W I for any set I ⊆ {1, . . . , l} such that I j I.

For (x, λ, µ) ∈ R n × R k × R l , we consider the mapping:

L(x, λ, µ) = ∇ρ(x) + k i=1 λ i ∇F i (x) + l j=1 µ j ∇G j (x), F (x), µ 1 G 1 (x), . . . , µ l G l (x) ,
and for (x, λ, µ) ∈ R n × R k × R l-j , j ∈ {0, . . . , l}, and I j = (i 1 , . . . , i l-j ), we introduce the 2 l following mappings:

L I j (x, λ, µ) = ∇ρ(x) + k i=1 λ i ∇F i (x) + l-j s=1 µ is ∇G is (x), F (x), µ i 1 G i 1 (x), . . . , µ i l-j G i l-j (x) . Lemma 2.2. Let p ∈ S ε I j . The point p is a critical point of ρ| W G (ε) if, and only if, there exists a unique (λ, µ) ∈ R n × R l such that L(p, λ, µ) = 0. Moreover, p is correct ⇐⇒ µ i = 0 ∀i ∈ I j and µ i = 0 otherwise. Proof. From [6] Lemma 1.2, we know that p is a critical point of ρ| S ε I j ⇐⇒ ∃!(λ, μ) ∈ R k × R l-j , L I j (p, λ, μ) = 0. Hence L(p, λ, µ) = 0 where µ = (µ 1 , . . . , µ l ) with µ i = 0 if i ∈ I j μi if i ∈ I j .
The converse is obvious. It is straightforward to prove the unicity of (λ, µ) . Now let us suppose that p is correct. As p ∈ S ε I j and L(p, λ, µ) = 0, we necessarily have µ i = 0 for all i ∈ I j and as p is correct, µ i = 0 for all i ∈ I j . Conversely, if µ i = 0 for all i ∈ I j and µ i = 0 for all i ∈ I j , then

∇ρ(p) + λ i ∇F (p) + i∈I j µ i ∇G i (p) = 0.
We see that p is a correct critical point of ρ| W G (ε) .

Lemma 2.3. Let p ∈ S ε I j be a correct critical point of ρ| W G (ε) and (λ, µ) ∈ R k × R l such that L(p, λ, µ) = 0. Then p is a correct critical point of ρ| W G (ε) pointing inward if, and only if, (-1) ε i µ i < 0 for all i ∈ I j .

Proof. From the previous lemma, p is a correct critical point of ρ| W G (ε) if and only if ∃!(λ, µ) such that L(p, λ, µ) = 0 with µ i = 0 for all i ∈ I j and µ i = 0 for all i ∈ I j . Denoting by Π the orthogonal projection on T p W , we have

Π(∇ρ(p)) = - l i=1 µ i Π(∇G i (p)) = - i∈I j µ i Π(∇G i (p)).
We know that for all i ∈ I j , Π((-1)

ε i ∇G i (p)) points inward W G (ε), so Π(∇ρ(p)) = i∈I j (-(-1) ε i µ i )Π((-1) ε i ∇G i (p))
points inward if and only if -(-1) ε i µ i > 0, i.e. (-1) ε i µ i < 0 for all i ∈ I j .

In the next lemma, we want to relate the sign of the dierential of L at one zero (p, λ, µ) of L to the Morse index at p of ρ restricted to the stratum that contains p. Lemma 2.4. Let p ∈ S ε I j be a correct critical point of ρ| W G (ε) and let (λ, µ) ∈ R k × R l be such that L(p, λ, µ) = 0. Then p is a Morse critical point if, and only if, det(DL(p, λ, µ)) = 0. Moreover, we get that

sign det(DL(p, λ, µ)) =   i∈I j sign(G i (p)) i∈I j sign(µ i )   (-1) τ (p)+k+l-j ,
where τ (p) is the Morse index of ρ| S ε I j at p.

Proof. Let (p, λ, µ) be a zero of L:

DL(p, λ, µ) =               ( ) B ∇G 1 (p) . . . ∇G l (p) t B (0) (0) µ t 1 ∇G 1 (p)
. . .

µ t l ∇G l (p) (0) G 1 (p)
. . .

G l (p)               , with B = (∇F 1 (p), . . . , ∇F k (p)) ∈ M n,k (R) and ( ) ∈ M n (R).
We know that p ∈ S ε I j so µ i = 0 ∀i ∈ I j and G i (p) = 0 ∀i ∈ I j , which we denote by I j = (i 1 , . . . , i l-j ). Then we get:

DL(p, λ, µ) =   i∈I j G i (p) i∈I j µ i                 ( ) B ∇G i 1 (p) . . . ∇G i l-j (p) t B (0) (0) t ∇G i 1 (p)
. . .

t ∇G i l-j (p) (0) (0)               , =   i∈I j G i (p) i∈I j µ i   DL I j (p, λ, μ),
where μ ∈ R l-j is such that μi = µ i ∀i ∈ I j .

From [START_REF] Szafraniec | The Euler characteristic of algebraic complete intersections[END_REF] Lemma 1.4, we know that p is Morse ⇐⇒ det(DL I j (p, λ, μ)) = 0, and we have

det (DL(p, λ, µ)) =   i∈I j G i (p) i∈I j µ i   det DL I j (p, λ, μ) . So det(DL(p, λ, µ)) = 0.
Finally, still from [START_REF] Szafraniec | The Euler characteristic of algebraic complete intersections[END_REF] Lemma 1.4, we know that

sign det DL I j (p, λ, μ) = (-1) τ (p) (-1) k+l-j , hence we obtain sign det (DL(p, λ, µ)) =   i∈I j sign(G i (p)) i∈I j sign(µ i )   (-1) τ (p)+k+l-j . Remark 2.5. If k + l = n and p is a correct critical point of ρ| S ε I 0 , its Morse index τ (p) is 0. Now we choose ρ(x) = ω(x) = 1 2 (x 2 1 + • • • + x 2
n ) so we have ∇ω(x) = x. For all positive constant R, we set:

B n+k+l R = {(x, λ, µ) ∈ R n × R k × R l | x 2 + λ 2 + µ 2 ≤ R 2 }, S n+k+l-1 R = {(x, λ, µ) ∈ R n × R k × R l | x 2 + λ 2 + µ 2 = R 2 }.
We would like to compute deg ∞ L, the topological degree at innity of

L L : S n+k+l-1 R →
S n+k+l-1 for some R > 0 but we need to prove rst that it is well dened.

Lemma 2.6. The set L -1 (0) is compact.

Proof. Let us study the vanishing set of L.

We easily nd that

L -1 (0) = j,I j (p, λ, µ) ∈ R n × R k × R l | (p, λ, μ) ∈ L -1 I j
(0) with μi = µ i ∀i ∈ I j and µ i = 0 otherwise , which we can also write

L -1 (0) = j,I j L -1 I j (0) × 0 j
(up to changes of indices in the coordinate system).

From [START_REF] Szafraniec | The Euler characteristic of algebraic complete intersections[END_REF] Lemma 2.1, the set L -1

I j
is compact for all j ∈ {0, . . . , l} and so is the product L -1 I j (0) × {0}. As the union on j and I j is nite, we can conclude that the set L -1 (0) is compact. Theorem 2.7. Let R > 0 be such that L -1 (0) B n+k+l R . If the sets W and G -1 i (0), i = 1, . . . , l satisfy the transversality condition ( ), then ε

(-1) |ε| χ(W G (ε)) = (-1) k deg ∞ L.
Proof. As in the proof of Theorem 2.2 in [START_REF] Szafraniec | The Euler characteristic of algebraic complete intersections[END_REF], we can choose a function ω : R n → R equal to ω on R n \ B R , such that ω| W and ω| W G (ε) , with ε ∈ {0, 1} l , have only Morse correct critical points, and nally such that ω uniformly approaches ω for the Whitney C 2 -topology.

For (x, λ, µ) ∈ R n × R k × R l , we consider L(x, λ, µ) = ∇ω(x) + i λ i ∇F i (x) + i µ i ∇G i (x), F (x), µ 1 G 1 (x), . . . , µ l G l (x) . As L -1 (0) ⊆ B n+k+l R , for ω close enough to ω, we also have that L-1 (0) ⊆ B n+k+l R . Let (x, λ, µ) ∈ L-1 (0).
Then there exists j ∈ {0, . . . , l} such that x ∈ W I j and from Lemma 2.2, x is a critical point of ω| W I j . But we have supposed that ω| W I j only had Morse correct critical points, so from Lemma 2.4, we know that det D L(x, λ, µ) = 0. Thus, 0 ∈ R n × R k × R l is a regular value of L. Moreover, the set L-1 (0) is nite. We now want to compute the degree of L, which is equal to the degree of L since if ω and ω are close enough, L L and L L are homotopic on a suciently big sphere.

Let us dene

C L = {(p, λ, µ) ∈ R n × R k × R l | L(p, λ, µ) = 0} the zero set of L.
Then we consider:

F I j = {µ ∈ R l | µ i = 0 ∀i ∈ I j , µ i = 0 ∀i ∈ I j }
and the following subset of C L:

C ε I j = (p, λ, µ) ∈ C L | µ ∈ F I j , (-1) ε i G i (p) > 0 if i ∈ I j , (-1) ε i +1 µ i > 0 if i ∈ I j .
From Morse theory for manifolds with corners ( [START_REF] Dutertre | On topological invariants associated with a polynomial with isolated critical points[END_REF]), for a given ε, we have Example 2.8.

χ(W G (ε)) = l j=0 I j p∈Πn(C ε I j ) (-1) τ (p) , where τ (p) is the Morse index of ω| W I j at p and Π n the projection of R n × R k × R l on R n . Hence ε (-1) |ε| χ(W G (ε)) = ε ( -
f (x 1 , x 2 ) = -(x 1 + 3) 3 x 2 -3x 2 1 + x 2 2 + x 1 , G(x 1 , x 2 ) = x 2 1 + x 2 2 -30.
Here, the transversality condition ( ) is satised. We have L : R 2+1+1 -→ R 2+1+1 (x, λ, µ) -→ (∇ω(x) + λ∇f (x) + µ∇G(x), f (x), µG(x)).

We get that deg ∞ L = -2 and thus ε∈{0,1}

(-1) |ε| χ(W G (ε)) = χ(f -1 (0) ∩ {G ≥ 0}) -χ(f -1 (0) ∩ {G ≤ 0}) = -deg ∞ L = 2.
The sets f -1 (0) ∩ {G ≥ 0} and f -1 (0) ∩ {G ≤ 0} are respectively four and two lines so are respectively homotopic to four and two points. Hence their respective Euler characteristic are 4 and 2. Thus, χ(f -1 (0) ∩ {G ≥ 0}) -χ(f -1 (0) ∩ {G ≤ 0}) = 4 -2 = 2.

(- 1 )(- 1 )

 11 1) |ε| l j=0 I j p∈Πn(C ε I j ) (-1) τ (p) . But for (p, λ, ∈ C ε I j , we know that signD L(p, λ, µ) ε i (-1) card(I j ) = (-1) |ε| (-1) l-j . 1) τ (p) (-1) k+l-j ,anddeg ∞ L = deg ∞ L,we can conclude that ε |ε| χ(W G (ε)) = (-1) k deg ∞ L.

Figure 1 .

 1 Figure 1. Curves G -1 (0) and f -1 (0).


Alberto Badías
email: abadias@unizar.es

Icíar Alfaro

David González

Francisco Chinesta
email: francisco.chinesta@ec-nantes.fr

Elías Cueto
email: ecueto@unizar.es

Reduced order modeling for physically-based augmented reality $

Keywords: Model order reduction, data assimilation, augmented reality

In this work we explore the possibilities of reduced order modeling for augmented reality applications. We consider parametric reduced order models based upon separate (affine) parametric dependence so as to speedup the associated data assimilation problems, which involve in a natural manner the minimization of a distance functional. The employ of reduced order methods allows for an important reduction in computational cost, thus allowing to comply with the stringent real time constraints of video streams, i.e., around 30 Hz. Examples are included that show the potential of the proposed technique in different situations.

Introduction

The so-called virtuality continuum is a continuous scale between reality and a complete virtual world. In between, every imaginable combination of real and virtual objects can be envisaged: real objects in a virtual environment, virtual objects in a real world or any other possible combination you can think of. In general, despite the plethora of different names that have been proposed to name different situations, augmented reality is usually considered as being placed somewhere in the middle of the continuum, see Fig. 1.

While Google usually refers to the continuum as immersive computing, Microsoft usually employs the term mixed reality.

Reality

Augmented Reality Virtual Reality In that sense, it seems that there is no consensus in the exact difference between terms like augmented reality, AR, or mixed reality, MR, for instance.

Instead, it is often preferred to see the different possibilities as different points in the continuum sketched in Fig. 1. To date, most of the existing AR applications embed static virtual characters in a real environment (think of the famous PokèmonGo app, for instance). However, what is really challenging is to be able to embed physically realistic virtual objects in a real environment (through video streams, glasses, etc.) and make them interact. The challenge comes precisely from the fact that modern video standards record at some 30 frames per second (fps). For instance, a modern smartphone usually offers the possibility to record at 30 or 60 fps. Therefore, to embed a virtual object in a video stream will imply to be able to generate synthetic images -compliant with the laws of physics-at those 30 fps so as to make it realistic.

In this work we aim at producing physically-based synthetic images at such rates. This could have a vast range of possible applications, from monitoring and decision making in the framework of industry 4.0, to assistance for laparoscopic surgery, to name but a few [START_REF] Haouchine | Monocular 3d reconstruction and augmentation of elastic surfaces with self-occlusion handling[END_REF]. For instance, augmented reality could allow us to project in different formats (such as tablets or smartphones, glasses, ...) hidden information within a manufacturing process (stresses, defects, ...). In laparoscopic surgery, on the other hand, critical information such as the precise location of blood vessels is often hidden for the surgeon [START_REF] Haouchine | Impact of soft tissue heterogeneity on augmented reality for liver surgery[END_REF]. In this work we are considering the possibility of enriching existing objects with hidden information as well as incorporating synthetic objects to the scene so as to help in the industrial design process, for instance. Projecting all this information on top of video streams could be of utmost interest for these and many other applications. Very few applications consider, on the other side of the virtuality continuum, the possibility of manipulating reality so as to make it appear as modified by virtual objects. The sole exceptions seems to be [START_REF] Davis | Image-space modal bases for plausible manipulation of objects in video[END_REF], where each video frame is considered as a two-dimensional elastic continuum that can be deformed according to the presence of a virtual object. Reality appears thus modified by the presence of virtual objects if wee it solely through the video stream.

If the information to be added to the video is to be extracted from numerical simulation, the difficulties are mainly two-fold. On one hand, there is the need to couple the computer model (usually obtained by means of finite element techniques) to the physical environment, through the lens of the camera. This process is known as registration. After registration, our method must be able to track the evolution of the physical environment so as to interact with it. If the environment is rigid and we have one single but moving camera, the problem is known to be well posed and to have a unique solution. This problem is called Structure from Motion, SfM, in the computer vision community [START_REF] Civera | Camera self-calibration for sequential bayesian structure from motion[END_REF]. It includes the problem of determining the pose -location and orientation-of the camera. If one or more objects in the real, physical scene is deformable, the problem turns out to be ill posed, although good solutions exist under different assumptions [START_REF] Hartley | Multiple View Geometry in computer vision[END_REF]. Non-Rigid Structure from Motion, NRSfM, [6] [7] and Shape from Template, SfT, methods [8] [9] are some good examples of different approaches to this problem.

On the other hand, our model should be able to generate results at the prescribed rate (as mentioned before, think of some 30 Hz). If we consider, for instance, deformable solids with general non-linear constitutive laws under finite strain settings, the problem of solving such a model some thirty times per second becomes an evident bottleneck. To avoid such problems, we are employing in this work reduced order models. These enable us to minimize the computational cost of the evaluation of the model while guaranteeing a prescribed level of accuracy. In fact, as will be noticed hereafter, the problem will be cast in the form of a classical data assimilation problem, for which the minimization of a functional will be needed. This minimization procedure is greatly beneficed from the separated (affine) parametric form of the unknown field of the model.

The precise type of reduced order model is not an essential ingredient of the technique here presented. For instance, Reduced Basis techniques [START_REF] Quarteroni | Certified reduced basis approximation for parametrized pde and applications[END_REF] [11] [START_REF] Maday | A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations[END_REF] have been employed successfully for data assimilation problems [START_REF] Manzoni | Accurate solution of bayesian inverse uncertainty quantification problems combining reduced basis methods and reduction error models[END_REF] [14] [START_REF] Manzoni | Computational reduction for parametrized pdes: Strategies and applications[END_REF]. Here, Proper Generalized Decompositions (PGD) have been employed [START_REF] Ladeveze | Nonlinear Computational Structural Mechanics[END_REF] [17] [START_REF] Ladeveze | The latin multiscale computational method and the proper generalized decomposition[END_REF] [19] [START_REF] Chinesta | Recent advances in the use of the Proper Generalized Decomposition for solving multidimensional models[END_REF] [21] [22] [23]. This technique constructs the solution of the problem in the form of a finite sum of separate functions. This particular form of the solution, which has a lot in common with Reduced Basis techniques, greatly simplifies the task of inverse identification that will be needed in this type of problems, as will become clear hereafter [START_REF] Gonzalez | Proper generalized decomposition based dynamic data driven inverse identification[END_REF] [25]

[26] [START_REF] Ch | Proper generalized decomposition based dynamic datadriven control of thermal processes[END_REF].

This paper is structured as follows. In Section 2 we describe the problem.

Particular attention is paid to the necessary link between the video stream and the reduced order model of a (possibly non-existent) deformable solid. In order to project a synthetic image on top of each frame under real-time constraints, it is first necessary to locate and orient the camera (in other words, to determine its pose). This will allow to track deformations in the scene, whose vast majority will be nevertheless rigid, in general. In Section 3 we will briefly review the essentials of Proper Generalized Decompositions and introduce how they will allow us to solve the resulting data assimilation problem advantageously. In Section 4 we show how the separate representation of the solution (affine parametrization) allows for an extreme simplification of the problem, thus allowing to obtain the mentioned feedback rates. In Section 5 we show three different examples of application of the proposed technique.

First, we consider a linear elastic example in which fiducial markers have been employed to determine the pose of the camera. The use of fiducial markers could be possible in industrial environments, for instance, but is strictly not permitted for laparoscopic surgery applications, on the contrary. In a second example we consider a hyperelastic (thus, non-linear) example without any fiducial marker. In a third example we consider the possibility of introducing a purely synthetic (non-existing) object in the scene. These three examples show the potential of the technique. The paper ends with a discussion and some research lines to be pursued.

Problem setting

Our vision of the problem of augmenting a video stream with information arising from real time simulation is that of a data assimilation problem.

Under this rationale, the "experimental measurement" phase of the problem consists in extracting information about the deformation of the solid(s) appearing in the video. In this section we review some of the essentials of computer vision that will help us to perform this task. Excellent books on the field are available for the interested reader, such as, for instance, [START_REF] Hartley | Multiple View Geometry in computer vision[END_REF] or [START_REF] Szeliski | Computer vision: algorithms and applications[END_REF].

Soon after this brief review on the essentials of computer vision is performed, we cast the problem in the form of a general data assimilation problem and introduce its main ingredients. Notably, the very stringent real-time requirements imposed by video streaming (ranging usually from some 30 to 60 or even more frames per second) poses the main difficulties for any simulation method.

Measurements: the non-rigid structure from motion problem

We assume that we are recording a scene with a standard, perspective projection video camera (RGBD cameras, for instance, are not considered in this work) [START_REF] Hartley | Multiple View Geometry in computer vision[END_REF]. This camera is producing, after calibration and lens distortion correction [START_REF] Bradski | Learning OpenCV: Computer vision with the OpenCV library[END_REF], as it moves, two-dimensional images of the three-dimensional environment. The passage from the three-dimensional world to the flat, twodimensional images is actually a perspective projection, which we will denote by Π. The Structure from Motion, SfM, problem consists in determining the three-dimensional structure of a rigid scene from a sequence of images taken from different points of view, see Fig. 2.

The input image space is denoted by I i ⊂ R 2 , with i = 1, . . . , n images , the number of available images. This number of images is in general growing continuously while we record with our camera, since a batch approach is in general not allowed in this work.

x 1 x 2 x 3 I 1 x t Ω t p Π z I n
The a priori unknown camera operator is here denoted by Π : ∂ * Ω t → I. This camera projection can be expressed as Π = K[R|t], where K corresponds to the camera intrinsic matrix and [R|t] represents the camera extrinsic matrix. By denoting x c t = (x, y, z) and p = (p 1 , p 2 ) we have,

⎡ ⎢ ⎢ ⎢ ⎣ p 1 p 2 p 3 ⎤ ⎥ ⎥ ⎥ ⎦ = λ ⎡ ⎢ ⎢ ⎢ ⎣ f / d x 0 c x 0 f / d y c y 0 0 1 ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎣ R t 0 1 ⎤ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ x y z 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
, where x c t represents the coordinates of x t in the camera system (a moving frame of reference attached to the camera, not to be confused with the world reference frame, depicted in Fig. 2). ∂ * Ω t denotes the visible part of the boundary of the object of interest, Ω, at time t. The camera extrinsic matrix transforms 3D scene points into 3D points in the camera reference. It establishes the rotation (R) and translation (t) between the corresponding points. The camera intrinsic matrix K contains information about the projection centre (c x ,c y ), the pixel size (d x ,d y ) and the focal distance (f ), which is used to map 3D camera points into 2D pixel coordinates. The whole equation makes possible the transformation of real 3D points into image pixels.

Finally, λ represents a scale factor.

Remark 1. Note that we have introduced two indexes, i and t, that may lead to confusion. i is purely discrete and refers to the frame number. On the contrary, t refers to time, and is therefore continuous. It will be subsequently discretized, of course, but even in this case, i and t need not to refer to the same quantity, nor the intervals Δt and [i, i+1] need to be of the same length.

Once discretized, frame i captures time instant t, and sometimes we employ them indistinctly in an abuse of notation, but these indexes should not be confused.

Note that, with one single image, there is no way to determine if the image in the picture, p, has been produced by a big object located far away from the camera or a smaller one located close to it. To locate both the camera (its pose: position and orientation) and a rigid object we need more than one single image. One of the key ingredients of the SfM problem is the minimization of the reprojection error, i.e., the geometric error corresponding to the image distance between a projected point and a measured one [START_REF] Hartley | Multiple View Geometry in computer vision[END_REF]. As sketched in Fig. 2, when a camera captures a picture of the environment, it is actually performing a projection of the 3d points in the form p = Π(x), with p ∈ R 2 and x ∈ R 3 (we omit for simplicity the superscript t). However, in practice we do not know exactly the true value of x. It can be estimated by using well-known computer vision techniques, so as to give an estimate x of the true 3d position of the point. The interested reader can consult [START_REF] Bradski | Learning OpenCV: Computer vision with the OpenCV library[END_REF] or [START_REF] Hartley | Multiple View Geometry in computer vision[END_REF], for instance. We therefore can think of a projection (re-projection, in fact) of this estimate, for each frame, p = Π(x), such that the reprojection error would be d(p, p). In this work we are employing different techniques to determine the pose of the camera and to construct a map of the environment. This is indeed an intended choice, just to show that the proposed method works for any SfM problem. In the final example we show how the proposed method does not need any fiducial marker, as it employs the ORB-SLAM algorithm developed in [START_REF] Mur-Artal | ORB-SLAM: A Versatile and Accurate Monocular SLAM System[END_REF].

When a deformable solid is located in the scene things become more intricate. A deforming solid is equivalent to having one single image p t per configuration x t = φ(X, t), and thus the problem becomes ill-posed. It is not possible, in general, to determine the three-dimensional geometry of a solid with one single image. However, since we take pictures every 1/30 seconds, every configuration in the sequence of images, despite not being identical, is indeed very similar to the previous ones. An impressive effort of research has been devoted to solving the so-called Non-Rigid Structure from Motion (NRSfM) problem. A non-exhaustive list of techniques in this field could consider the Shape from Template (SfT) techniques [START_REF] Bartoli | Shapefrom-template[END_REF] [9], an Extended Kalman Filter approach employing linear elastic plate finite element to simulate the visible surface of the solid [START_REF] Agudo | Real-time 3d reconstruction of non-rigid shapes with a single moving camera[END_REF] [6], or recent low-rank approaches to the that initialize the sequence with rigid motion only and sequentially improve the basis of the method [START_REF] Paladini | Sequential Non-Rigid Structure-from-Motion with the 3D-Implicit Low-Rank Shape Model[END_REF], to name a few.

Assimilation of the data

Our approach to the problem will be that of deterministic data assimilation [33] [24]. Even if it is possible to cast the problem in a Bayesian framework (see, for instance, some recent references that employ different types of Kalman filters in the context of reduced order modeling, [START_REF] Marchand | Real-time updating of structural mechanics models using kalman filtering, modified constitutive relation error, and proper generalized decomposition[END_REF] [35]

[36]) we assume that observations coming from video streams are exact. Measurements coming from the NRSfM problem are indeed displacements of tose points identified as belonging to deformable media. This constitutes a set of punctual displacements u * .

Under this rationale, we consider the parametric dependence of the solution on a set of parameters μ ∈ D ⊂ R n param . These parameters will represent some hidden information to the user, such as the position of a load, for instance, that may be not visible in any of the frames. From the video stream, any of the aforementioned NRSfM techniques will provide a set of measurements -typically, displacements at some particular positions in the solidby minimizing the reprojection error, for instance. These will be denoted as u * (μ). The problem will be given by the following two discretized PDEs:

State equation: A(u, μ)u(x, μ) = f (μ), Observation equation: u * (μ) = C(μ)u(x, μ).
(

) 1
We highlight here the fact that the stiffness matrix A(u, μ) could depend (possibly in a non-linear manner) both on the unknown field u (usually, the displacement field) and the parameters μ. u ∈ R n FOM represents the set of nodal (finite element full order model) degrees of freedom and u * ∈ R n obs represents the set of observations taken from the video stream. The observation matrix C is assumed to be linear and is usually simply Boolean and μ-independent.

The deterministic approach to this (inverse) problem could thus be cast in the following form:

μ * = arg min μ∈R n param 1 2 u * -C(μ)u(x, μ) 2 2 , (2)
i.e., we seek to minimize the discrepancy between the observations u * and the model predictions u(μ) at particular locations given by matrix C. Frequently, this formulation is regularized by employing Tikhonov methods [START_REF] Manzoni | A reduced-order strategy for solving inverse bayesian shape identification problems in physiological flows[END_REF],

but in this case we have not found it necessary.

The computational complexity of problem (2) prevents nowadays computers to obtain the desired feedback response at 30 Hz. This is so, at least, for state-of-the-art (possibly non-linear) models involving hundreds of thousands (if not millions) of degrees of freedom. This is at the origin of the need for some type of model order reduction. In this work, we have employed Proper Generalized Decompositions, whose basics are reviewed next. This will allow us to avoid the usage of high dimensional vectors u ∈ R n FOM , but to substitute them by surrogate reduced order approximations u ∈ R n ROM , with n ROM n FOM .

A brief review of Proper Generalized Decomposition

We will not go into every detail concerning Proper Generalized Decompositions, given the vast corps of literature available since the initial proposition of the method in [START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[END_REF] and its subsequent identification with the strategy also proposed by P. Ladeveze as an ingredient of his famous LaTIn method [START_REF] Ladeveze | Nonlinear Computational Structural Mechanics[END_REF].

The interested reader can consult recent books on the methods such as [START_REF] Chinesta | PGD-Based Modeling of Materials, Structures and Processes[END_REF] [19] [START_REF]Separated Representations and PGD-Based Model Reduction[END_REF] or review papers such as [39] [20].

The essential ingredient in PGD methods is the assumption that the solution can be expressed as a finite sum of separate functions or, in other words, that the field u(x, μ) admits an affine parametric dependence, i.e.,

u(x, μ) ≈

n modes i=1 F i (x) • G 1 i (μ 1 ) • . . . • G n param i (μ n param ), (3)
where the symbol "•" represents the Hadamard or Schur component-wise product. Functions F i , G j i are also known as "modes" to highlight their strong link to the modes employed in Proper Orthogonal Decompositionbased model order reduction methods. To determine their precise expression, Eq. ( 3) is substituted into the weak form of the problem at hand, from which Eq. (1.a) emanates. Let us assume that the strong form of the problem is: find u(x,μ) in an appropriate functional space such that

L(u) = f (x),
equipped with suitable boundary conditions. Here, L represents the differential operator acting on the essential variable, u(x, μ). The precise form of this operator will be made clear for each of the examples in Section 5.

To obtain the weak form of the problem, we multiply both sides by a test function and integrate over the high-dimensional domain,

Ω x ×Ω 1 ×...×Ω n param u * (L(u)-f )dx dμ 1 • • • dμ n param = 0 ∀u * ∈ H 1 0 (Ω x )×L 2 (μ). (4)
The separated form assumed for u, when substituted into Eq. (5.1), allows to perform integration in a separated manner. Of course, for this to be possible, the operator L must satisfy some separability assumptions as well.

We refer the interested reader to [START_REF] Modesto | Proper generalized decomposition for parameterized helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation[END_REF] for a detailed analysis on the separability of differential operators, or to [START_REF] Niroomandi | Model order reduction in hyperelasticity: a proper generalized decomposition approach[END_REF] in a hyperelastic context. We thus proceed by assuming that a rank-n approximation has just been constructed,

u n (x, μ) = n i=1 F i (x) • G 1 i (μ 1 ) • . . . • G n param i (μ n param ),
and look for a rank-1 improvement,

u n+1 (x, μ) = u n (x, μ) + F n+1 (x) • G 1 n+1 (μ 1 ) • . . . • G n param n+1 (μ n param ).
In this same spirit, the test function will look

u * (x, μ) = F * n+1 (x) • G 1 n+1 (μ 1 ) • . . . • G n param n+1 (μ n param ) + F n+1 (x) • (G 1 n+1 ) * (μ 1 ) • . . . • G n param n+1 (μ n param ) + . . . + F n+1 (x) • G 1 n+1 (μ 1 ) • . . . • (G n param n+1 ) * (μ n param ).
Once substituted into Eq. (5.1), the assumed separated form of the unknown and the test function allow to perform the integration also in a separated way.

The PGD algorithm is thus comprised by a greedy loop to determine the approximating sum plus a fixed-point, alternated directions scheme to find each of the modes. The interested reader could consult any of the mentioned references for further details and, in particular, [START_REF] Cueto | Proper Generalized Decompositions: An Introduction to Computer Implementation with Matlab[END_REF] or [START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations[END_REF] for detailed Matlab implementations.

In order to determine the number of modes in the approximation, n, a stopping criterion is mandatory. Error estimators have been proposed for PGD approximations that could be employed to this end [START_REF] Ammar | An error estimator for separated representations of highly multidimensional models[END_REF] [44] [START_REF] Moitinho De Almeida | A basis for bounding the errors of proper generalised decomposition solutions in solid mechanics[END_REF].

PGD-based data assimilation

The procedure previously reviewed allows us to substitute u in Eq. (1.a)

by its just computed rank-n approximation. Since this approximation is computed off-line and once for life, the savings in computational cost is usually of several orders of magnitude. Once a reduced-order approximation to u has been determined, the minimization in Eq. ( 2) is performed here by employing the well-known Levemberg-Marquardt method [START_REF] Ammar | Parametric solutions involving geometry: A step towards efficient shape optimization[END_REF] [46] [START_REF] Ch | Proper generalized decomposition based dynamic datadriven control of thermal processes[END_REF]. In this method, one has to determine the sensibilities of the solution with respect to the optimization parameters, i.e.,

∂u ∂μ j (x, μ) ≈ n i=1 F i (x) • G 1 i (μ 1 ) • . . . • ∂G j i (μ j ) ∂μ j • . . . • G n param i (μ n param ).
Thanks to the separated (affine) structure of the parametric dependence, sensibilities could also be computed off-line and stored in memory so as to be particularized during the runtime procedure. The reader can notice the extremely well suited structure of PGD approximations for such type of inverse problems.

In what follows we introduce three different examples in linear and nonlinear elasticity that show the performance of the method and its suitability for augmented reality applications.

Examples

A linear elastic example employing fiducial markers

The first example is a linear elastic cantilever beam subjected to a punctual load applied at a variable position s, see the sketch in Fig. 3.

F i (x)G i (s)
was computed. In this case the classical elasticity problem is assumed to govern the physics. Its weak form consists in finding the displacement u ∈

H 1 (Ω) × L 2 (Γ t2 ) such that for all u * ∈ H 1 0 × L 2 (Γ t2 ): Γ Ω ∇ s u * : σdΩdΓ = Γ Γ t2 u * • tdΓdΓ
where Γ = Γ u ∪ Γ t represents the boundary of the beam, divided into essential and natural regions, and where Γ t = Γ t1 ∪ Γ t2 , i.e., regions of homogeneous and non-homogeneous, respectively, natural boundary conditions.

Here, t = P • δ(xs)e k , where δ represents the Dirac-delta function and e k the unit vector along the z-coordinate axis (we consider here, for the ease of exposition, a load P directed towards the negative z axis of reference).

Once regularized, the Dirac-delta term is approximated by a truncated series of separable functions in the spirit of the PGD method, i.e.,

t j ≈ m i=1 f i j (x)g i j (s)
where m represents the order of truncation and f i j , g i j represent the j-th component of vectorial functions in space and boundary position, respectively.

For this particular example, only j = k (i.e., the third component of the vector) is assumed to exist.

In this example, fiducial markers were employed to help locate the deformed configuration of the beam. See Fig. 4 for a picture of the experimental device. The use of fiducial markers is particularly advantageous for some industrial applications (structural health monitoring, for instance), for which there is no limitation to the presence of small adhered paper or plastic pieces, for instance. If this is not possible -for aesthetic reasons, for instance, or just because we have no a priori access to the studied solid-feature-based approaches must be preferred. These will be analyzed in Section 5.3 below.

The number of modes n employed in this example was 19, even if for this problem much less are enough -see the big relative decay in modulus of the modes-. The first four spatial modes are represented in Fig. 5, while modes on s are depicted in Fig. 6.

Results of the real-time assimilation procedure are reported in Fig. 7, where different snapshots of the raw video, assimilated load position and augmented video are reported. Quantitative results are included in Table 1.

Note that the reported errors are absolute and represent the difference between the assimilated load position and the actual one. Except for load positions very close to the clamped support, that produce almost no deflection in the beam, and therefore a large error in the assimilation procedure, errors are less than 1% for most positions (for instance, 0.3% error for the load at s = 460 mm, or 0.1% for s = 420 mm). These values can be improved by employing more PGD modes, on one side, but are strongly limited by the camera's resolution and the proximity of the beam to the objective. In other words, it is the portion of the solid covered by a single pixel that usually determines the level of precision. The resulting video can be consulted at https://youtu.be/BK0sHfvjITo.

In it, the robustness of the proposed technique to the partial occlusion of some of the fiducial markers or even a total occlusion of the camera are tested. The system resulted to be very robust and recovered immediately the load position without any problem.

A hyperelastic example without fiducials

A 797.5 × 200 × 105 mm 3 foam beam was considered in this example.

The beam was meshed from a picture of a deformed configuration under self-weight load, see Fig. 8.

An experimental campaign was accomplished so as to determine that its (homogenized) constitutive law could be assimilated to a Kirchhoff-Saint Venant law with E = 0.11 N/mm 2 and Poisson coefficient ν = 0.2. The Kirchhoff-Saint Venant model is characterized by the energy density functional given by

Ψ = λ 2 (tr(E)) 2 + μE : E,
where λ and μ are Lame's constants and E the Green-Lagrange strain tensor.

In this case, the weak form of the problem, from which Eq. (1.a) emanates, takes the form

Γ Ω(t) E * : C : EdΩdΓ = Γ Γ t2 u * • tdΓdΓ,
where represents the fourth-order constitutive tensor. Given the non-linear strain measure E, the resulting problem is inherently non-linear.

Despite the well-known limitations of the Kirchoff-Saint Venant law under compressive stresses, this law showed to fit much better than usual hyperelastic laws (neo-Hookean, Mooney-Rivlin) to the experiments. In fact, some mild viscoelastic effects were observed, that are responsible of some oscillations in the results -see the accompanying video-, but these were nevertheless neglected. For short-term periods, results showed to be in good accordance to this law, as will be detailed hereafter.

A mesh of 18938 nodes and 98863 linear tetrahedral elements was created.

Boundary conditions were assimilated to a pinned support (foam-wood contact surface showed to be nearly an adhesion) on the leftmost 40 mm. On the right, a sliding support was accomplished by means of plastic laminates.

15 N loads were applied by means of lead disks with 80 mm diameter, whose size was considered in detail in the simulation (due to the weak stiffness of the foam, punctual loads provoked very poor results by greatly deforming the upper surface of the foam).

The goal of this experiment was to develop augmented reality videos under real-time constraints (no batch processing was allowed) in which the position of the load was determined and the displacement field was plotted on top of the actual geometry of the beam, so as to indicate the user its true magnitude. Therefore, the displacement field was -as in Section 5.1assumed to have a parametric form

u = u(x, s),
where x ∈ Ω represents the coordinates of the considered point and s the position of the applied load along the upper face of the beam. A PGD approximation of this parametric displacement field of the type

u(x, s) ≈ n i=1 F i (x)G i (s)
was constructed off-line. n = 19 modes were considered in this approximation, although some experiments with only 11 modes provided results with very similar levels of accuracy. The load position coordinate, s was assumed to be one-dimensional, since the load was placed centered on the upper surface of the beam. The mesh along this dimension comprised 60 nodes in the central part of the beam (a portion of 100 mm of the beam at the right and left ends was not considered, since the displacements provoked by the load in these regions was not perceptible, due to the absence of bending).

A detailed derivation of the necessary linearization of the Kirchhoff-Saint Venant law under the PGD framework was accomplished in [START_REF] Niroomandi | Real-time simulation of biological soft tissues: a PGD[END_REF]. The reader is referred to this reference or to [START_REF] Cueto | Proper Generalized Decompositions: An Introduction to Computer Implementation with Matlab[END_REF] for a complete Matlab implementation.

Figure 9 shows the first five G i (s) modes of the model. Figure 10 shows modes F i (x), i = 1, 2, 3 and 11. Figure 11 shows a frame (second 55 of the video) in which we appreciate the accuracy of the assimilated displacement field and, notably, the location of the load position, marked with an arrow. The video was recorded at a resolution of 1280 × 720 p. (which is the resolution of an iPhone 6). Under these circumstances, the proposed method is able to provide a response at 60 Hz. The mean error is on the order of 1.5 mm. This value is also dependent on the relative proximity of the Note that no fiducial marker is employed at this stage.

Neo-Hookean hyperelasticity

We consider in this example the process of design and analysis of an automotive boot sealing made of a neo-Hookean rubber. The geometry of the seal is shown in Fig. 12. The model is composed by a mesh of 8640 nodes and 5940 linear hexahedral elements. The rubber is assumed to follow an incompressible law

W = C 1 (I 1 -3),
where W represents the strain energy density function, C 1 = 1166 MPa for this particular example, and I 1 represents the first invariant of the right Cauchy-Green strain tensor. The material is thus considered incompressible.

The lever is assumed to be rigid, and rotates around a pinned support in the plane where the boot seal lies.

In this particular example the boot seal does not exist physically in the video. Instead, this example opens the possibility of employing augmented reality for engineering design purposes, by showing the performance and appearance that the virtual object could have in the physical world. In this particular example, the displacement field of the boot seal is parameterized by the angle of rotation of the lever, θ. As in the previous examples, we hypothesize that this dependence can be affinely approximated as

u(x, θ) ≈ n i=1 F i (x)G i (θ).
(5)

Construction of the reduced order model.

For this particular example, the so-called non-intrusive PGD method has been employed [49] [50]. In particular, we have employed the formulation introduced in [50], that employs adaptive samplings of the parametric domain and collocation approaches. We thus avoid complex linearization procedures while employing off-the-shelf commercial softwares to construct the sought PGD approximation, Eq. ( 5). Some of the spatial modes F i (x) are shown in Fig. 13. The parametric space θ ∈ [0 • , 52 • ] -here, symmetry has been employedhas been discretized into 193 finite elements. Probably much less degrees of freedom could be necessary but, given the inherent non-linearity of the problem due to the presence of contact phenomena, a very fine discretization was employed that nevertheless did not prevent the method to reach the real-time constraints.

Segmentation and registration.

In this example, as already mentioned repeatedly, there is no physical boot seal. Therefore, the objective of the computer vision implementation is to determine the degree of rotation θ of the lever. To that end, no fiducial markers have been employed. In the absence of that markers, computer vision systems employ feature matching algorithms, capable of detecting and tracking local frame features. In this work we employ ORB feature detectors [51]. These are the green markers in Fig. 14. Within SLAM algorithms, ORB detectors provide with a very efficient means of unveiling the environment in which we move [START_REF] Mur-Artal | ORB-SLAM: A Versatile and Accurate Monocular SLAM System[END_REF]. SLAM methods were primarily designed for robot navigation in rigid environments. But in this particular example the objective is to detect the lever tilt angle θ. In other words: there is some part of the scene that is not rigid. To detect θ an algorithm is proposed in which we proceed by parts. First of all, it is necessary to detect the plate in which the lever is mounted. To that end, we employ Random Sample Consensus (RANSAC) methods [52]. RANSAC methods allow to fit a model from experimental data in the presence of a big number of outliers. This is precisely our case: a significant portion of the ORB features lie on a plane (precisely the plate in which the lever is mounted), while the rest of the detected features in the environment are considered outliers. The resulting detection is shown in Fig. 15, where the detected plate is depicted by means of red features, while the environment is depicted in blue. In Fig. 16 the results of the ORB-SLAM [START_REF] Mur-Artal | ORB-SLAM: A Versatile and Accurate Monocular SLAM System[END_REF] reconstruction of the environment, together with the found pose of the camera (blue squares) are shown. This method allows to track the movement of the camera in a very accurate manner. Once the plate has been detected, the tilting angle of the lever is found by filtering the white color of the lever from the images. This allows for a very precise determination of the angle θ, see Fig.

Performance.

The resulting video from which frames in Fig. 17 have been extracted

can be seen at https://youtu.be/0whA92vFk1g. In it, a robustness test is made for the proposed method in which the camera is moved away of the lever position, so that it disappears from the image. The localization capabilities of ORB-SLAM methods are able, nevertheless, to find the lever back with great precision, thus closing the loop and restarting the augmentation with no problems.

Discussion

In this paper a new method for the physically realistic augmentation of video streams through numerical simulation is proposed. The method, given the stringent constraints imposed by modern video frame rates (30 or 60 frames per second in an iPhone, for instance) is based on the employ of reduced order modeling and simulation. In particular, the affine (separate) parametric approximation of most reduced order techniques is exploited for a very efficient data assimilation procedure that enables us to obtain a very accurate registration between reality and our model.

The proposed method is valid for any augmented reality application in which we are interested about hidden information in the video governed by a PDE (such as stress, for instance), but also to create and augment with objects not present in the physical reality. In any case, state-of-the-art computer vision techniques have been employed so as to detect the camera pose and to map the environment. Three different examples have been analyzed

in which we employ different approaches to the problem of simultaneous localization and mapping of the environment. In the first of them we employed fiducial markers, a very versatile possibility in industrial settings, where we can mark the object of interest with such badges. In the second one we employed some existing pattern in the solid so as to track deformations in it, while in the third example we employed feature-based approaches (in particular, ORB-SLAM) to locate the solid, track its deformation and the camera pose.

In any of the three different examples, the usage of reduced order methods (here, Proper Generalized Decompositions) allowed for a very efficient and accurate determination of the variables of interest at usual video frame rates.

A standard iPhone 6 camera was employed for every example.

The presented method opens the door for more complex augmented reality applications that truly enable a linkage of simulation techniques with video formats, thus widening the possibilities of modern CAE techniques.

Varona and Eduardo Estopiñán, whose help with the foam example is greatly appreciated.

Figure 1 :

 1 Figure 1: The virtuality continuum.

Figure 2 :

 2 Figure 2: Sketch of the Structure from Motion problem.

Figure 3 :

 3 Figure 3: Cantilever beam problem. A moving load P is parameterized through its position coordinate s.

Figure 4 :

 4 Figure 4: Experimental device for the aluminum beam example. Note the fiducial markers for the location of the beam and camera pose determination (big four circles) and the markers (small blue dots adhered to the beam) for the determination of the beam deflection.

Figure 5 :Figure 6 :

 56 Figure 5: Modes F i (x), i = 1, 2, 3, 4 for the aluminum beam example. These are vectorial modes. The colour map corresponds to the vertical displacement, the most relevant one.

Figure 7 :

 7 Figure 7: Three different snapshot of the raw video (left column), simulation (center) and augmented video (right). The colour map corresponds to the modulus of the displacement field.

Figure 8 :

 8 Figure 8: Geometry of the foam beam under self-weight load. This image was taken as the reference configuration of the beam. Fiducial markers were employed at this stage, that were later eliminated, to determine the pose of the camera and help to construct the reference configuration of the beam.

Figure 9 :

 9 Figure 9: Modes 1, 2, 3 and 11on s, the position of the load coordinate. Note that, as it is the case very often with PGD, the modes increase in frequency, the first one having the lowest frequency content.

Figure 10 :

 10 Figure 10: Modes F i (x), i = 1, 2, 3 and 11 for the foam beam example. Again, modes increase in frequency as i increases. This a typical PGD result, although in some, very rare cases, modes show a somewhat more irregular pattern, see [48], for instance.

Figure 11 :

 11 Figure 11: Frame of the resulting video (second 55) in which we appreciate the assimilated position of the load (see the arrow). Superimposed, the displacement field of the beam.

Figure 12 :

 12 Figure 12: Geometry of the rubbery boot seal (left) and a snapshot during the process (right). The angle θ is the fundamental magnitude to be determined by the computer vision system, since the boot seal is entirely virtual. It does not exist in the physical reality.

Figure 13 :

 13 Figure 13: Different modes F i (x), for the boot seal non intrusive approximation.

Figure 14 :

 14 Figure 14: Raw video with the detected features in green. Note that there is no boot seal in the physical reality.

Figure 15 :

 15 Figure 15: Detection by RANSAC methods of the plate where the lever is located (in red) and reconstruction of the rigid environment (in blue).

17

 17

  .

Figure 16 :

 16 Figure 16: Results of the ORB-SLAM tracking of the camera pose and reconstruction of the environment [30]. Red points represent the detected features, blue squares represent the came pose at different time instants and the green lines indicate common found structures between frames.

Figure 17 :

 17 Figure 17: Two different augmented frames of the video. Color map corresponds to the modulus of the displacement field at the plane of symmetry (right) and von Mises stresses (left).

Table 1 :

 1 Experimental results for the cantilever beam problem. Actual load position versus assimilated one. Errors are absolute.

	Actual	Mean	Difference	Actual	Mean	Difference
	Position	Estimated	abs. value	Position	Estimated	abs. value
	s (mm)	Pos. (mm)	s (mm)	(mm)	Pos. (mm)	(mm)
	0	0.14	0.14	260	262.21	2.21
	20	7.98	12.02	280	281.10	1.10
	40	26.64	13.36	300	301.30	1.30
	60	57.66	2.34	320	320.44	0.44
	80	82.09	2.09	340	338.33	1.67
	100	102.54	2.54	360	358.57	1.43
	120	122.67	2.67	380	383.49	3.49
	140	143.89	3.89	400	401.35	1.35
	160	164.55	4.55	420	420.70	0.70
	180	182.25	2.25	440	440.27	0.27
	200	204.27	4.27	460	458.40	1.60
	220	222.75	2.75	480	477.69	2.31
	240	242.22	2.22

Acknowledgements

We are grateful to our colleague at the Aragon Institute of Engineering Research, prof. J. M. M. Montiel, whose guidance through the world of computer vision has been crucial to us. We are also grateful to Miguel Ángel

$ This work has been supported by the Spanish Ministry of Economy and Competitiveness through Grants number DPI2017-85139-C2-1-R and DPI2015-72365-EXP and by the Regional Government of Aragon and the European Social Fund, research group T88.

approach. International Journal for Numerical Methods in Biomedical Engineering, 29(5):586-600, 2013.

[48] Siamak Niroomandi, Iciar Alfaro, Elias Cueto, and Francisco Chinesta.

Model order reduction for hyperelastic materials. International Journal for Numerical Methods in Engineering, 81(9):1180-1206, 2010.