Sylvie Fainzang
email: sylvie.fainzang@orange.fr

ANTHROPOLOGY, ETHICAL DISSONANCE AND THE CONSTRUCTION OF THE OBJECT

In this article, I discuss certain questions relating to the ethical difficulties faced by anthropologists when dealing with two different social groups and when one group holds a position of dominance over the other. In the first example, I draw on my work on doctor-patient relationships in France; in the second, on a study on reproduction in immigrant African families from Mali and Senegal, living in polygynous households in France. I use these examples to explore questions of positionality, beneficence and potential harm. I show the choices I made in order to construct an epistemologically ethical object.

The importance of ethical considerations in research is part of the contemporary anthropological landscape [START_REF] Faubion | An Anthropology of Ethics[END_REF][START_REF] Fluehr-Lobban | Ethics and the Profession of Anthropology: Dialogue for Ethically Conscious Practice[END_REF][START_REF] Lambek | Ordinary Ethics: Anthropology, Language, and Action[END_REF][START_REF] Caplan | The Ethics of Anthropology: Debates and Dilemma[END_REF], particularly in medical anthropology (Anthropologies et sociétés, 2000;Scheper-Hughes, 2005;Singer and Bauer, 2007, who stressed the need for political engagement). For example, a special issue of American Ethnologist (2006) was devoted to the difficulties faced by researchers because of the bureaucratic ethics regulations managed by Institutional Review Boards (notably [START_REF] Katz | Ethical escape routes for underground ethnographers[END_REF]Shweder, 2006;[START_REF] Fassin | The end of ethnography as collateral damage of ethical regulation?[END_REF][START_REF] Lederman | The perils of working at home: IRB "mission creep" as context and content for an ethnography of disciplinary knowledges[END_REF]. The place of ethics in ethnographic research, and the epistemological principles and ethical concerns that are involved, have also been examined in medical anthropology, in a special issue of Ethnographiques.org [START_REF] Desclaux | Introduction au dossier 'L'éthique en anthropologie de la santé : conflits, pratiques, valeur heuristique[END_REF]. In this, the contributors addressed the problems posed by applying the ethical principles of medical research to the practice of ethnography in relation to issues such as informed consent and confidentiality, and examined the dilemmas researchers had to deal with on the ground (for instance, [START_REF] Beaudevin | Inaugurer" l'anthropologie de la santé au Sultanat d'Oman[END_REF][START_REF] Musso | A propos du "malaise éthique" du chercheur : les leçons d'un terrain sur les objets « sida » et « immigration » en France[END_REF][START_REF] Sarradon-Eck | Médecin et anthropologue, médecin contre anthropologue : dilemmes éthiques pour ethnographes en situation clinique[END_REF][START_REF] Desclaux | Les lieux du "véritable travail éthique" en anthropologie de la santé : terrain, comités, espaces de réflexion ? ethnographiques[END_REF]Sakoyan, 2008).

To further explore the place of ethics in ethnographic research on health and suffering, in this article I focus on the difficulties posed when studying a question which involves two distinct but interrelated social groups, whether they are in an antagonistic relationship or an inclusive one. I use two examples from my field work to discuss this. The first concerns my research into doctor-patient relationships in urban hospitals in France [START_REF] Fainzang | Patient Information between Public Space and Anthropology: Ethnography's Contribution to the debate[END_REF], which led me to examine the practice of lying; below I explain my decision not to address this from an ethical perspective. I show that the ethical stance for anthropologists consists of analyzing lying as any other social practice, without making an a priori moral assessment of the practice. In the second example, I draw on a difficulty I encountered during a study of reproduction in immigrant African families from Mali and Senegal [START_REF] Fainzang | La femme de mon mari[END_REF]. In these polygynous households in France, male domination is particularly pronounced and women are led to behave either submissively or rebelliously. I examine the difficult situation in which the defence of certain values to the benefit of one group may be simultaneously seen as detrimental to the population to which it belongs. In such circumstances, anthropologists find themselves having to make choices (Appell, 1978;[START_REF] Cassell | Ethical Problems of Fieldwork[END_REF]. These choices depend on each anthropologist's sensibility and his/her ideological, philosophical, or political opinions. I describe this situation as 'ethical dissonance', in reference to the notion I coined to account for the difficulties doctors face, when confronted with differing or even opposing values through which they must navigate [START_REF] Fainzang | La relation médecins-malades: information et mensonge[END_REF].

In this discussion, I refer to the primordial ethical principle specified in the AAA Statement of Ethics (2012) that one must 'do no harm'. 1 The common factor of these two situations is the focus on two groups of people linked by a relationship of power: one the doctor/patient relationship, the other relations between the sexes. In each I explain the nature of the ethical problem and how I managed it, since, as [START_REF] Fassin | L'éthique au-delà de la règle. Réflexions autour d'une enquête sur les soins en Afrique du Sud[END_REF] has highlighted, ethics should be perceived as a process of construction during which researchers make the problems they encounter explicit, explore the stakes involved, and negotiate the possible responses. In the two cases, the question I had to address concerned the construction of the object of research, in connection with the reactions that these types of research provoked in the medical environment for the first and in the academic world for the second. I explore methodological and ethnographic dimensions of these studies in order to allow the reader to better grasp the issues discussed here.

AN ETHNOGRAPHY OF LYING: BETWEEN ETHICS AND EPISTEMOLOGY

The first case I examine is a study on the provision of information and lying between doctors and patients in France. The study was mainly conducted in two departments (oncology and internal medicine) of two different urban hospitals, and aimed to reveal the logics and mechanisms underlying information exchange between doctors and patients. Despite the variability between departments and their different specialities, similar behavior in terms of patient information can be observed within the services studied. As Van der Geest and Kinkler (2004) argued, "Hospitals both reflect and reinforce dominant social and cultural processes of their societies" (1995). The study included 80 patients, 60 with cancer and 20 with chronic inflammatory or autoimmune diseases. I met the patients at different stages of illness, with some considered as virtually cured and only coming to hospital for check-ups, while others were receiving treatment (chemotherapy, radiotherapy or surgery) or were under palliative care. They were of different ages (from 30 to 80, with a majority between 55 and 70), of both sexes and from various socio-professional backgrounds.

The investigation consisted of observing medical consultations and then separately meeting the doctors and the patients to see how the verbal exchange was constructed and to decode the reasons for and mechanisms of their actions and words. I used open-ended interviews with patients, paying attention to the questions that they asked themselves, the questions they asked the medical personnel, the answers they were given, the information they were provided, and their reactions. I also conducted non-directive interviews with 12 doctors to learn how they understood the issue of information and how they perceived patient expectations. No formal ethical clearance needed to be obtained to conduct this work, but in addition to getting the consent of the doctors and patients, I simply had to request authorization from heads of the medical departments.

The objective of this study was not only to gather data on how doctors give patients information, but also on how patients pass on information to doctors, and on the lies they sometimes tell each other. The research was thus based on two distinct orders of potentially antagonistic thought, produced by patients and doctors respectively. To achieve this objective, it was important not to favor one point of view over the other and not to take sides, but to have what I have described elsewhere as "dual empathy" (Fainzang, 2010: 102), that is to say, empathy for both sides. This dual empathy allowed me to highlight the social issues at stake. Indeed, in seeking knowledge, 2 anthropologists must strive to develop empathy with the Other, but also, in studying social relationships, with 'the Other of the Other' (doctors versus patients, the strong versus the weak), even if one may feel better able to identify with some groups than others. Thus it is appropriate to distinguish between what I call 'moral empathy' and 'methodological empathy', that is, between the predisposition to put oneself in the shoes of the group studied, and the need to deliberately try to do so. For [START_REF] Engelen | Current Disciplinary and Interdisciplinary Debates on Empathy[END_REF], empathy combines aspects of thinking and feeling. In distinguishing between understanding the other cognitively and understanding the other emotionally, they refer to the cognitive and affective dimensions at work within empathy (whether or not these can be separated). In this vein, my definition of moral empathy and methodological empathy aims, on one hand, to indicate the postures that allow one to arrive at these two forms of understanding, and on the other hand, to highlight what distinguishes them: moral empathy refers to the anthropologist's tendency to put him/herself in the place of those who s/he initially has a greater ability to understand emotionally, whereas methodological empathy refers to the willingness or attempt to understand, at least cognitively, the reasons for the actions of the people under study, without necessarily conforming with them. Understanding the emotional state of another person need not entail sharing it.

Most works carried out on the question of truth in medicine, mainly medical, philosophical or sociological works [START_REF] Abiven | Mentir pour bien faire? ou le mensonge en médicine[END_REF][START_REF] Barnes | A Pack of Lies: Towards a Sociology of Lying[END_REF][START_REF] Bok | Lying: Moral Choice in Public and Private Life[END_REF][START_REF] Constant | Tout le monde n'a pas droit à la vérité[END_REF][START_REF] Delaporte | Dire la vérité au malade[END_REF][START_REF] Fletcher | Morals and Medicine. The moral Problems of the patient's right to know the truth[END_REF][START_REF] Geets | Vérité et mensonge dans la relation au malade[END_REF][START_REF] Hoerni | Paroles et silences du médecin[END_REF][START_REF] Kuran | Private Truths, Public Lies. The social consequences of Preference Falsification[END_REF][START_REF] Shibles | Lying: A Critical Analysis[END_REF], either take the form of justifying what is referred to as 'therapeutic privilege' or else make pleas for patient participation. Because the question of truth and lies is often addressed in ethical terms, many authors choose it as their subject in order to defend a particular position, while underlining the ethical dilemma of whether to tell the truth or lie. Some follow the Hippocratic oath which states that one must do a patient no harm, and they interpret this as a reason to reveal to patients their true condition or prognosis; others take this as a reason to shield patients from such information that concerns them. As we can see, 'do no harm' is in itself a very subjective judgment. When the debate is framed in ethical terms, partisans and adversaries of truth construct their points of view and elaborate their justifications on the same grounds: the same arguments ('for the good of the patient' or 'to do no harm') are used to support radically opposing positions. Additionally, those who advocate patient information are also divided between those who support information in itself (the patient's right to know) and information for a determined end, as a means of convincing patients of the need to follow a certain treatment.

In the first case, it is a question of informing the patient as a person with the right to this information concerning his/her health; in the second, it is a question of giving information with the aim of obtaining better treatment compliance. The first position supports a moral position, a principle, while the second takes a utilitarian position, equivalent to a therapeutic strategy. The positions that actors develop combine ethical and therapeutic options in order to justify informing a patient or not (or even lying to them), but these are often relatively arbitrary. The arbitrary nature of these positions is reinforced by the fact that doctors often find themselves caught between opposing choices. I qualify this situation as 'ethical dissonance' to account for the contradictory principles and opposing values through which doctors must navigate, and to underline the disharmony between the different ethical principles to which they have to refer. They must on the one hand reassure the patient (a role that some attribute to the paternalistic model of the doctor-patient relationship), and on the other, empower the patient, allowing them their liberty to manage their own life and their body (a role which fits into a more egalitarian relationship). Taking sides in this debate would have meant opting for one principle over another, and risked obscuring the true stakes involved in the mechanisms underlying the practices of actors. But, while it was relevant to study the conditions and meanings of social practices developed in relation to ethical principles, my aim was not to take a stance in this ethical debate.

It was therefore necessary to tackle the issues in a different way from how they are generally approached. I had to look at the subject from an anthropological perspective, and to analyze the situation without inferring any normative position. This approach required a certain number of epistemological conditions. Firstly, it required that, from an emic point of view, I start not with my conception of what a lie is, but with that which the actor believes to be true or not. This then involved accepting the purposely iconoclastic nature of the term 'lie' as applied to the medical profession, with no judgment on the practice of lying from the onset. Though some doctors prefer to use the terms 'withholding information' or even 'lying by omission', others advocate using the term 'lie', whether in the medical literature on the subject [START_REF] Hoerni | Pratiques et axiologies de l'anthropologie face à la domination politique[END_REF][START_REF] Abiven | Mentir pour bien faire? ou le mensonge en médicine[END_REF] or in the field. Thus, for example, a doctor had told his patient, who had just had a rectal operation, that he had a polyp rather than cancer. The patient said: "I must say you sent me for a big operation for just a polyp!" The doctor answered: "Well, to be honest, it wasn't a polyp." The patient said: "You mean it was cancer?" The doctor (embarrassed): "It was a pre-cancerous lesion which would have become a cancer in a few weeks." Later, he told me: "In fact, that was a lie too since the 'pre' in pre-cancerous was false. He really did have cancer". This echoes Dr Merran's words as quoted by [START_REF] Favereau | Le silence des médecins[END_REF]: "Honestly, we often end up telling patients barefaced lies" (24).

The negative connotation of the term 'lie', used to describe a practice carried out by doctors, is often lessened by the fact that the so-called 'medical' lie is justified by a discourse that rationalizes it and provides legitimacy, even morality. Plato wrote in La République (1966) that only doctors and chiefs of cities possessed the right to lie, the former in the interest of their patients and the latter in the interest of the city. The lie told by a doctor is thus rationalized, legitimized, and even ethically justified. On this subject, [START_REF] Barnes | A Pack of Lies: Towards a Sociology of Lying[END_REF] highlights the existence of 'permissible' lies and shows that the manuals on medical ethics often include a section on lies that can be told to patients if they are dying, if they are receiving placebos, or if they are taking part in clinical trials. However, using the term 'doctors' lies' during interviews upset some of them, even those who admitted to lying themselves. They thought that the word damaged their image 3 and undermined their position of power over patients (their 'therapeutic privilege'). It is as if, given the negative moral burden of the word, only doctors possess the right to refer to their practice as lying, and as if their perception of the ethically legitimacy of doing so prevents non-doctors from qualifying the practice in these terms.

Furthermore, examining lying as a bilateral practice made it possible to shift focus and to look at the therapeutic relationship as a social relationship. I then had to try to 'understand' the doctors' and the patients' reasons for acting, their motivations, the mechanisms of this information, and the evolution of doctors' and patients' social roles. This was a prerequisite to avoiding the risk of either becoming a mere spokesperson for the patients (which is not the researcher's role) or subjecting the study to a biomedical problematic by reducing the debate to one of the ethical stances of the health care personnel.

The inquiry revealed that while doctors think they give information preferentially to those they deem psychologically capable of dealing with it, in fact information is largely given to people who appear to belong to more privileged social backgrounds, that is, to higher social categories. Doctors prejudge their patients' capacity to understand the information they are provided with and their ability to accept it (in an amalgam between these two types of ability) based on external signs (language, clothing, body posture), approximate clues to social or cultural capital. By selecting those who receive information, doctors help to reinforce social inequalities in access to information.

When anthropologists examine a social question embedded in the public arena, they have to do so using their own epistemological stance. I was able to demonstrate the discriminations that characterize the modalities of information provision to patients, only because my objective was not to provide a moral or ethical perspective of the practice, but to move away from the usual stance (whether psychological, therapeutic or ethical) in order to define the cultural and social mechanisms involved. This made it possible for me to produce new results likely to enrich understanding of the social issues related to lying. This relates to the connection between ethical posture and construction of the object. Ethics and epistemology are strongly connected. For [START_REF] Lederman | Ethics: Practices, Principles, and Comparative Perspectives[END_REF], the link between the two can be seen in the tactical use of deception concerning the aim of the research. For me, however, attempting to approach the subject with a perspective that was not focused on the ethical issue was the sine qua non condition to understand the context of these practices and the social issues at stakealthough, of course, a complete withdrawal from ethical commitment is not possible since the anthropologist's own emotional positions remain part of the disciplinary approach, as [START_REF] Crapanzano | Réflexions sur une anthropologie des émotions[END_REF] and [START_REF] Luhrmann | Subjectivity[END_REF] have shown. 4 My point here is that one can choose not to address a question from an ethical perspective for ethical, methodological and epistemological reasons. This involves not setting a research question a priori according to the values to which the anthropologist adheres or to the judgments which s/he might be tempted to make. The ethical stance for anthropologists consists then of analyzing lying as any other social practice, without making any moral assessment of this practice, whatever their posture or personal opinions, even if, for example, their convictions lead them to deplore medical paternalism and the inadequacy of patient information. Lying can be studied as an aspect of social life, even if it involves revealing the role it plays in exercising or resisting power. Likewise, one can study the underlying reasons that doctors choose to lie to their patients in a non-judgmental manner even if one hopes to work to promote better patient information. There are, of course, many shades of grey in between 'truth' and a 'lie', with halftruths, dissimulation, deliberate misunderstandings, and so on, as I have analyzed earlier [START_REF] Fainzang | La relation médecins-malades: information et mensonge[END_REF].

The ethical character of the research resides in approaching the question with a perspective which is not ethical, both to decipher the arbitrariness of ethical positions on the subject and to be able to understand the social and anthropological reasons for the practices under examination. Here, to be epistemologically ethical is to approach the construction of one's object in a markedly different way from an ethicist, that is, on one hand, to withhold judgment and, on the other hand, to look beyond the usual psychological explanations, based on ethical arguments, in order to try to understand the social mechanisms at the root of the provision (or not) of information to patients.

ETHNOGRAPHY OF POLYGYNY: AN ETHICAL SUBJECT?

My second example comes from research I carried out some 25 years ago among West-African immigrant families (in particular the Soninke and Toucouleur ethnic groups) living in polygynous marriages in France. The object of the study was to understand how this type of household is organized, how the women manage their bodies, their sexuality and their health, particularly in relation to reproduction, and the relationships formed between men and women in the household. The research, permitted by the husbands and warmly welcomed by the wives, aimed to observe the relationships between husbands and wives on one hand and between cowives on the other, and to study their impact on how the families manage biological reproduction. Although the husbands immediately assigned me a physical place with the women in adherence to the sexual division of domestic space, even when a family lived in a single room, my investigations also included men and their perceptions of the social relationships within their household.

The polygynous institution represents a particular modality of socializing feminine sexuality.. It responds to imperatives and choices that are political, economic, and cultural [START_REF] Fainzang | La femme de mon mari[END_REF]. The marital sexuality of women is subject to a number of limitations and prohibitions that leave no doubt as to its primary vocation: procreation. However, procreation is heavily codified, and sexual relations within a polygynous household are subject to precise rules.

Although polygyny is authorized and ratified by Islam with a limit of four wives, and is legitimized by the state in some countries where Islam is the dominant religion, the institution of polygyny is pre-Islamic. The legitimacy of the institution goes hand in hand with the establishment of strict principles dictating that the husband should treat each co-wife equally. This principle is best expressed by the institutionalized practice of 'turns', which regulates domestic and sexual obligationthese two obligations are generally linked together. In reality, women are often subject to unequal treatment and there is often strong rivalry between cowives, particularly in relation to sexuality and reproduction. They traditionally take the form of competing for reproductive work. As a vector of prestige and recognition, biological reproduction is a fundamental concern for women in polygynous societies, determining their social and marital status, since the more children a woman has, the more prestige she has, as a wife, in the eyes of the husband and the whole community. Rivalries between women also result in accusations of maraboutage, persecutory magical practices destined to jeopardize the other's reproductive capacity, and usually used to explain the onset of sterility and other sexual and reproductive health problems.

However, with immigration to a society where polygyny is not the norm, polygynous families may face economic and cultural realities that induce some women to want to interrupt reproduction and access contraception. When their husband refuses, these women sometimes take birth control pills secretly or adopt the strategy of resisting sexual serviceswhat women refer to as a 'sex strike'at the risk of compromising, they think, the future of their children because of the belief that a disobedient wife puts the health and morality of their progeny in danger. This weapon may be more or less effective depending on the woman's position in the household, since the possibility of real resistance is limited to the women most sexually valued by their husbands. Women resist their husband's will by using their sex as a weapon. In doing so, they reproduce the role assigned to them but also the social definition by which they are defined as women, since this weapon constrains them in an image and a definition of feminity, which renders them sexual and reproductive objects.

Some of my male anthropologist colleagues were critical of my choice of research subject because they believed that it risked harming the 'African population in France'. In their eyes, studying questions such as 'African polygyny' posed an ethical problem, since highlighting a practice of discrimination or domination would contribute to stigmatizing those who sanction it. I would therefore be contributing to the marginalization of the community with which I wished to work. While their criticism did not deter me from the project, it certainly weighed on my conscience, for their position suggested I should have judged the subject to be taboo and not even contemplated it. Yet the contemporary anthropological view is that culture is not a homogenous whole, and that differences of power among individuals within the group should be examined and exposed [START_REF] Massé | Les limites d'une approche essentialiste des ethnoéthiques. Pour un relativisme éthique critique[END_REF]Gupta and Fergusson, 1997;Hours et Selim, 2000). Anthropologists therefore also have an ethical obligation to emphasize the different forms of power that exist within a social group, including showing how those who hold a role of dominance in one context may be dominated by others. Polygyny is one form of domination, and not writing about it, I felt, perpetuated this. I believed that it was important to attempt to understand the symbolic logics and social pressures that govern relationships between men and women in the context of polygyny, particularly so because numerous immigrant women I had met had complained to me about their lives and asked me to take an interest, as an anthropologist, in their fate, in order to make their situation known.

So I chose to undertake this research, fully aware that this choice was linked to ethical and political options. At the same time, consistent with the principle of methodological empathy discussed above, I also decided to consider the social constraints to which the men were subject. Men were under considerable pressure to become polygynous and experienced great difficulty in avoiding this, with compatriots repeatedly cite the Soninke adage, "having only one wife, is like having only one shirt."

As this case indicates, anthropologists face difficulty in deciding between the possible choices available to them when working on a subject may be beneficial but, at the same time, may risk doing harm. The issue here is not concerning cultural relativism, which made it possible to formulate "a critical challenge specifically to Western values and beliefs" (Lederman, 2012: 4) though, as a backlash to this, it is with increasing reference to Western values that ethical reflection is developing today in anthropology. Within such a framework, one of the ethical questions could have been: How much should we silently acknowledge situations that go against the values we hold [START_REF] Cefai | L'engagement ethnographique[END_REF]? And how far may our ethnographic practice contribute to endorsing or showing indifference to these situations? As Fluehr-Lobban (2013) and [START_REF] Lambek | Ordinary Ethics: Anthropology, Language, and Action[END_REF] have suggested, the ethical debate must extract itself from one on relativism, so that the question of ethics is not reduced to debates over cultural relativism versus ethical relativism. Well beyond the debate on cultural relativism, there lies another question, of the ethical conflicts that arise from respecting epistemological norms.

In the case examined here, the conflict arose between two possible research postures when faced with the object under study. These postures are difficult to reconcile when a collision or even a contradiction occurs between the different social and political implications of the study, that is, when working on one specific issue could be seen both as a heuristic necessity or as a means of improving the overall situation of the people being studied (namely, the women who are subject to polygynous marriage), and as a risk of doing harm to the image of the wider group to which the subjects belong ('African' families). Here, the anthropologist also finds him/herself in a situation of 'ethical dissonance' to the extent that, in attempting to meet one ethical principle (help the weak, do good), s/he may fail to meet another (do no harm). This notion seems to me also appropriate in characterizing the difficulties that anthropologists face when they agree with two conflicting value registers. Ethical dissonance forces anthropologists to make choices that are necessarily arbitrary, since they depend on their own ideological, philosophical or political options. The notion of 'ethical dissonance' here accounts for the contradictory ethical principles to which the anthropologist adheres, and which s/he has to arbitrate. While what is described as an ethical dilemma by [START_REF] Appel | Ethical Dilemmas in Anthropological Inquiry: A Case Book[END_REF] is intended to characterize the problems encountered by anthropologists in cross-cultural inquiry, with an emphasis on the mental conflict between ethical imperatives as experienced by the anthropologist, the concept of ethical dissonance aims to underline the very contradiction between the various principles to which s/he adheres.

DISCUSSION

I am not dealing here with a conflict between two heterogeneous cultural systems or value systems, or even with the problem posed by the fact that the inquiry can take place at the intersection between divergent interests, as discussed by [START_REF] Appel | Ethical Dilemmas in Anthropological Inquiry: A Case Book[END_REF]. Appel underlined the irreconcilable character of the anthropologist's multiple roles and the various 'moral expectations' implied. In turn, [START_REF] Lederman | Ethics: Practices, Principles, and Comparative Perspectives[END_REF] has discussed the potential conflict between different 'reference frames', including professional expectations, personal choices and values, and the restrictions associated with the networks to which one belongs. In this context, she discussed the difficulty researchers face in reconciling their own point of view and that of their donors. This difficulty was referred to explicitly in the AAA Executive Board Statement on Ethics (2012) thus: "Anthropologists must weigh competing ethical obligations to research participants, students, professional colleagues, employers and funders, among others, while recognizing that obligations to research participants are usually primary" (9). There, the 'competing' ethical obligations are the different social obligations to which anthropologists are subject, towards their different social partners. Consequently, when the AAA discusses the risk of conflicting ethics, it is in reference to the ethical principles with which anthropologists are confronted as researchers, members of an association, or fathers, for example, that is, while they are wearing several hats.

The point I raise here is different; I am concerned with the choice that anthropologists need to make between two differing value registers, both of which they can agree with. An ethical posture and an engaged posture do not necessarily merge. Despite the obvious necessity of creating an ethical framework for all research, there are inherent difficulties within the ethical choices made during any given research project and the ethical and epistemological stakes they represent. Although an epistemological-ethical approach to a group first implies accounting for the social relationships that can be observed there, this is far from simple when doing no harm to one group might involve harming another.

The question, therefore, is whether defending certain values, as a foundation of the choice to carry out research on certain people or groups, could actually go against the good or the interest of others. Through the critical perspective they adopt, some anthropologists may aim to improve the lot of human beingsfor Nancy Scheper-Hughes in reference to her fieldwork in South Africa, it is even a "tool for human liberation " (1995: 415). But anthropologists have to confront questions related to the plurality of the people they study. The position I advocate is that we must free ourselves from the view that ethical posture should only be imagined in a dualistic and Manichean way, opposing good and bad. Anthropologists working in the field of medical are not necessarily facing nasty people (who discriminate, dominate others or put others' rights to well-being in danger) on one side, and good people on the other side. Anthropologists find themselves dealing with individuals who occupy socially distinct positions and interact with one another (men versus women, doctors versus patients, and so on).

Choosing to conduct research on polygyny in order to bring to light a situation of domination does not necessarily involve taking the women's side in terms of the analysis, no more than choosing to study patient information in order to highlight its inadequacy involves not trying to understand the feelings and motivations of doctors. In the case of polygynous households, researching the social practices at work also implied not taking the side of one spouse or another, although there were also relationships of domination between co-wives. Relations between the sexes are socially constructed, of course, and sexual roles are imposed as much on one sex as the other. Men are as socialized as the women, even if the schemas within which this socialization is constructed are radically different. The socialization of men also supposes forms of constraint. Proof of this can be found in the lack of regard accorded to monogamous men and the pressure placed on men to take on additional wives.

Developing a critical analysis of the processes that take place in the field of illness, rethinking the question of power, working to enable people to manage their bodies and their health in an autonomous manner, or studying situations of vulnerability, domination, and inequality [START_REF] Fainzang | Health, State and Politics (Medical Anthropology, Welfare State and Political Engagement I)[END_REF] does not protect a researcher from having to make choices about how the object is constructed and to review his or her position in the field. The two situations I have described have their own specific stakes. But both raise in their own ways the issue of constructing the anthropological object when this object is perceived or understood by civil society in ethical terms. In the first example, some health professionals reproached me for discussing doctors' lies because of the negative moral connotation of this practice; in the second example it was my colleagues who were worried about me discussing the functioning of polygamy in African immigrant families because of the negative judgment of this practice in Western societies. The two situations also illustrate that to be ethical, one's decisions should necessarily also be epistemological. To this aim, while the task of the anthropologist is to take interest in the Other, he/she must not neglect to examine the points of view of the 'Other of the Other'. Studying doctors and patients in the therapeutic relationship, or studying husbands and wives in polygynous households, can allow for greater complexity because the dominant parties have also been given a voice and have had their reasons heard. In this respect, the two issues raised here have in common an emphasis on the ethical difficulty faced by anthropologists when dealing with two different social groups in relationships of opposition or of inclusion. This leads us to examine formal ethics statements, such as those of the AAA, in a critical manner.

The Ethical Principles and Guidelines for the Protection of Human Subjects of Research, known as the Belmont Report (1979), requires researchers to conform to three ethical of Respect, Beneficence, and Justice (cf. [START_REF] Beauchamp | Ethical Issues in Social Science Research[END_REF][START_REF] Cefai | L'engagement ethnographique[END_REF]. However, simply respecting these principles does not exhaust the ethical questions raised in anthropological research. In line with the Belmont Report's recommendations, the Statement on Ethics established by the AAA Executive Board (2012), a reference point in the French anthropological community, lays down the Principles of Professional Responsibility. This latter text states that the primary ethical obligation is to do no harm and that "determinations regarding what is in the best interests of others or what kinds of efforts are appropriate to increase well-being are value-laden" (2012: 4). Yet, this assumes we are able to determine with certainty the good and the harm that our work can do. Is studying doctor's lies likely to harm the doctors, or does it benefit the patients? Further, this instruction conveys a slightly Manichean vision of what 'doing good' or 'avoiding doing bad' is, because the individuals and groups at stake are diverse and occupy non-identical social positions: Husbands are in a position of dominance in relation to their wives by virtue of the social construction of gender relationships in this framework, for instance, but they are in a position of vulnerability in French society by virtue of their social, economic and ethnic status as African migrants.

CONCLUSION

We can find ourselves confronted with problems routinely because of our choice of objects and how they are constructed. The question here was not about knowing if one should judge what one sees or nota question [START_REF] Scheper-Hughes | The Primacy of the Ethical: Propositions for a Militant Anthropology[END_REF] answered by imagining what forms a politically committed and morally engaged anthropology might takeor even whether it is possible to do so, although we could consider that judgment made at the beginning of the research is epistemologically prejudiced, but judgment at the conclusion of the research is ethically desirable. Anthropologists are confronted with numerous ethical questions in their fieldwork (Cassell and Jacobs 1987), whether they work beyond or within their national borders. Among those questions, the main one is generally that of knowing how to behave to do good or do no harm. As I have stressed, we should avoid approaching the issue in a Manichean way, all the more so because we sometimes work with people who may be both in a position of vulnerability and a position of dominance. Reflecting on the ethical questions that anthropologists face supposes highlighting the complexity of social postures occupied by the actors.

If the choices that anthropologists make are to satisfy both ethical and epistemological requirements, they need to be made at the intersections of these two types of requirement. This meeting point will be different for each researcher, and his or her responses to them will depend on their own sensibility to the phenomena of discrimination, injustice and power. Despite ethical codes such as the AAA Code of Ethics, 1998, anthropologists constantly find themselves having to manage new ethical questions in the course of their research. Discussing the problems they face and the choices they make are important to contribute to the collective thinking about ethical challenges in medical anthropology. NOTES 1. These three principles (of respect, beneficence, and justice) are divided up differently in biomedical ethics, where there are four principles, as the principle of non-maleficence is added to the principle of beneficence [START_REF] Beauchamp | Principles of Biomedical Ethics[END_REF]. 2. On the issue of empathy inherent in ethnographic practice, and on the distinctive character of anthropological empathy compared to medical empathy in relation to knowledge, see [START_REF] Fainzang | Anthropology and Medicine: Empathy, Experience and Knowledge[END_REF]. 3. On how the doctor profession is represented and the harm that the social sciences are accused of causing to their image, see [START_REF] Sarradon-Eck | Médecin et anthropologue, médecin contre anthropologue : dilemmes éthiques pour ethnographes en situation clinique[END_REF]. 4. In this respect, it is difficult not to react to the sometimes brutal words of some doctors, who, under the pretext that they are obliged to do so by law, bluntly disclose to a patient their diagnosis, or verbally abuse them. Not all doctors spare their patients' feelings, and the message can sometimes be very harsh.


Keywords: assimilation de données, problèmes mal posés, régularisation, assimilation d'images, flot optique, mouvement

Résolution des problèmes mal posés du traitement d'image par Assimilation de Données Résumé : L'assimilation de données est un cadre mathématique utilisé en science de l'environnement pour améliorer les prévisions météorologiques, océanographiques ou pour la qualité de l'air. Elle nécessite la résolution d'un système d'équations : une première décrit l'évolution temporelle d'un vecteur d'état, une seconde décrit le couplage entre l'observation et ce vecteur d'état et une dernière définit la condition initiale du vecteur d'état. Dans ce rapport, l'assimilation de données est utilisée pour résoudre un problème mal posé en traitement de l'image, habituellement résolus par des techniques de régularisation spatiotemporelle. Une approche générique est décrite pour résoudre le problème mal posé dans le cadre de l'assimilation de données. Cette méthode est illustrée sur le problème de l'estimation du flot optique. Un avantage est de pouvoir utiliser un critère de qualité des observations pour pondérer leur contribution des observations dans le calcul de la solution. L'équation d'évolution permet alors d'obtenir une régularité temporelle pertinente pour la solution.

Introduction

In the research field of Image Processing, most problems are ill-posed in the sense that it is not possible to provide a unique solution [START_REF] Hadamard | Lecture on Cauchy's Problem in Linear Partial Differen-Equations[END_REF]. A first cause of ill-posedness is that the equations used to model image properties are under-determined. An example is given by the famous "aperture problem" occurring in the estimation of optical flow: a further constraint is required to compute a unique field of velocity vectors. As an image processing problem is usually modelled by a system of equations to be solved, the so-called Image Model, this type of ill-posedness means that the Image Model is not invertible. A second cause of ill-posedness occurs when the computation of image features can be obtained by different algorithms. For example, determining image gradient requires to approximate a differential operator by a discrete one among several possible finite difference formulations; each one providing a different result.

A common strategy to solve ill-posed problems is to provide the Image Model with additional information. Two options may be considered. 1) Providing explicit information: additional images are used to enlarge the set of input data. However, this is generally not possible because other acquisitions having the requested properties are not available. 2) Providing implicit information: either hypotheses on image properties or constraints on the solution can be used. A usual constraint is to restrict the dimension of the space of admissible solutions. For instance, the result may be searched among the functions with bounded spatial variations: this is called a "Tikhonov regularization method" in the literature [START_REF] Tikhonov | Regularization of incorrectly posed problems[END_REF]. In the general case, these additional image properties or constraints are expressed as equations, which, combined to the Image Model, lead to a new invertible Image Model.

Assuming an image sequence is available, enhancement may be obtained by taking into account the temporal information. Let us illustrate this on image segmentation, which is a pure spatial problem. A spatial regularization method, such as Shah-Mumford's functional [START_REF] Inria Mumford | Optimal approximations by piecewise smooth functions and associated variational problems[END_REF], produces a segmentation which is a compromise between a smooth solution and the fidelity to the input data. The segmentation process can be performed directly on the whole sequence and not on a frame-by-frame basis: following [START_REF] Weickert | Variational optic flow computation with a spatio-temporal smoothness constraint[END_REF], the solution is seek as a function depending on the spatial and temporal coordinates. This approach has however several drawbacks. First, it imposes an arbitrary temporal regularity which can not deal with complex dynamics. Second, missing data are taken into account in the process and introduce errors in the final solution. By "missing data", we refer to pixels' values displaying a wrong information due to either a failure of the acquisition system or noise. Third, as the solution is looked for in the spatio-temporal domain, this leads to an high computing complexity compared to a pure spatial model, which is a limiting factor for operational applications.

An alternative to this spatio-temporal approach, particularly interesting in case of applications with a high probability of missing data, is to consider the temporal evolution of images over the sequence. The challenge becomes to write an effective and pertinent dynamics model and to include it in the solution computation. Such information can be inferred, for instance, from a priori knowledge on images and on the observed phenomena. Moreover, data quality has also to be evaluated in order to ignore missing data in the process.

In this paper, we propose to use the Data Assimilation framework as a generic tool to solve ill-posed Image Processing problems. A Data Assimilation method solves a system of three components with respect to a state vector: an evolution equation describes the evolution of the state vector over time, using an operator called the "evolution model";

an "observation equation" models the link between the state vector and the observations provided by the image sequence; the initial condition of the state vector.

Each component is inaccurate and a description of the error is stated in terms of a Gaussian noise characterized by a covariance matrix. This matrix depicts the dependencies existing between the components of the state vector on the one hand, and between two different locations in the space-time domain on the other hand.

As it will be proven in the paper, Data Assimilation is a suitable approach to solve the three drawbacks of Weickert's method because: the evolution model describes the images' temporal dynamics; the covariance associated to the observation equation error overcomes the problem of missing data by weighting, for each pixel, the importance of the observation equation in the computation of the solution; Data Assimilation is a frame-by-frame process which allows large image sequences to be processed.

This article is organized as follow. Section 2 introduces the concept and difficulties of ill-posed problems in the Image Processing research field. We give some typical examples and present a short start-of-the-art of Tikhonov regularization methods. Section 3 describes the variational Data Assimilation method known as the 4D-Var algorithm. How Data Assimilation can be used to solve ill-posed problems by assimilating image data within an appropriate evolution model is explained in Section 4. Section 5 is then a direct application and describes how to compute optical flow in this framework. Section 5 also presents and discusses experimental results. We conclude in Section 6 and give some scientific perspectives to this research study.

Ill-posed problems in Image Processing

Hadamard gives the following definition: a problem is well-posed if 1) it has an unique solution, 2) the solution depends continuously on the input [START_REF] Hadamard | Lecture on Cauchy's Problem in Linear Partial Differen-Equations[END_REF]. A problem which does not meet these conditions is called ill-posed. Using this definition, optical flow estimation, image registration, curves or surfaces matching, tracking of multiple objects, segmentation, restoration, deconvolution, denoising and shape from shading are ill-posed problems because the equations used for modeling these problems are under-constrained (the aperture problem of optical flow for instance).

Links between image properties and the solution are modeled as a set of equations constituting the Image Model. The image, input data, is denoted Y and depends on the spatial coordinate x in a bounded domain denoted Ω. The solution of the problem, denoted X, is not necessarily an image: it can be a velocity field, a curve, etc. To be general, the Image Model is described by:

Á(X, Y)(x) = 0 ∀x ∈ Ω (1)
INRIA with Á a differentiable image operator that may be: linear: Á(X, Y) = Y -A(X) with A linear. This is a typically the case in segmentation, restoration, denoising, deconvolution: Á measures the discrepancy between the input image and the solution filtered by the operator A.

non linear. A common situation in image processing is the following:

Á(X, Y) = B(Y(C(X)).
The input data Y is only evaluated on the pixels C(X) depending on the solution. For example, Á(X, Y)(x) = -∇Y(X(x)) for the well known active contours or Á(X,

Y)(x) = Y 2 (x+ X(x)) -Y 1 (x)
for estimating optical flow between two frames Y 1 and Y 2 .

In this section, attention is focused on variational methods: instead of directly solving (1), an optimization problem is formulated and the solution is 1) is under-constrained, a possible method to obtain a unique solution is to use a Tikhonov regularization. This is performed by adding a second term to the functional E which becomes:

obtained by minimizing E(X) = Ω Ψ( Á(X, Y) )dx with Ψ a convex function such as Ψ(0) = 0. If Equation (
E(X) = Ω   Ψ( Á(X, Y) ) + n≥0 α n Ψ ∂ n X ∂x n   dx (2)
An usual choice is to set α i = 0 for i = 1 ensuring a first order regularization or α i = 0 for i = 1, 2 ensuring a second order regularization. The norm used in the regularizing term is often the Euclidean norm but other choices are possible such as X 2 = X T AX, A being a symmetric and definite positive matrix. It is even possible to choose A depending on the input image Y in order to perform a regularization driven by Y. For instance, to determine optical flow, Nagel uses in [START_REF] Nagel | Displacement vectors derived from second-order intensity variations in image sequences[END_REF]) a matrix A depending on the local image configuration: the algorithm performs a weak regularization on edges and strong otherwise.

The minimization of E is led in the calculus of variation framework: the solution is searched as the zero of the Euler-Lagrange equation associated to (2), ∂E ∂X = 0, with ∂E ∂X denoting the differential of E with respect to X. For a first order regularization, the general expression of this Euler-Lagrange equation is:

-∇. Ψ ′ ( ∇X ) ∇X ∇X + 1 α 1 ∇. Ψ ′ ( Á(X, Y) ) ∂Á ∂X = 0
It is discretized by finite differences and the solution is obtained using a Jacobi or Gauss-Seidel method in the linear case and a method of steepest descent or conjugate gradient otherwise.

From the beginning of this section, the image is considered as only depending on the spatial coordinate and the Image Model is purely spatial. In the case of optical flow computation, at least two frames are required, but the sequence is still processed on a frame by frame basis. As a consequence, the result obtained on one image has no link with those obtained on adjacent ones. An improved solution, when dealing with a temporal sequence, has been proposed by Weickert [START_REF] Weickert | Variational optic flow computation with a spatio-temporal smoothness constraint[END_REF] and consists in minimizing the functional:

E(X) = Ω T 0   Ψ( Å(X, Y) ) + n≥0 α n Ψ ∂ n X ∂x n , ∂X ∂t   dxdt (3)
with X and Y becoming time dependent and the regularization term computed in the space-time domain. Such a functional describes accurately regular dynamics but becomes irrelevant for non linear ones. Moreover, missing data are taken into account: the method can only perform a smoothing of aberrant values using the spatio-temporal neighborhood. Last, the time is viewed as an additional dimension and a direct consequence is the proportional increase of the problem's size. A huge memory and computer time is then requested for processing a large sequence.

A partial answer to the problem of dealing with complex temporal dynamics and missing data is to add a model of the dynamic evolution. This implies to be able to solve simultaneously the Image Model and the evolution model and this can be achieved using the Data Assimilation framework described in the following Section.

3 The Data Assimilation framework

Modelisation of the problem

The Data Assimilation framework aims to solve the system (4,5,6) with respect to a state vector X(x, t), depending on the spatial coordinate x and time t:

∂X ∂t (x, t) + Å(X)(x, t) = E m (x, t) (4) X(x, 0) = X b (x) + E b (x) (5) Y(x, t) = À(X)(x, t) + E O (x, t) (6)
Equation ( 4) describes the temporal evolution of X. Å, called evolution model, is supposed differentiable. As Å may describe approximately the evolution of the state vector, a model error E m is introduced to quantify the imperfections. Equation (5) establishes the initial condition of the state vector and the error is expressed by the background error E b . Equation ( 6), called observation equation, describes the links between the observation vector, Y(x, t), and the state vector.

The observation may sometimes be a part of the state vector, in which case À is a projection operator. In other cases, the observation may be an indirect function of the state vector. Equation ( 6) is the standard form of the observation equation used in the Data Assimilation literature. However, this formulation is quite restrictive to describe the links, possibly complex, existing between the observation and the state vector. For instance, non-linear Image Models can not be expressed in this form. To be more general, the following will be used in this article:

À(Y, X)(x, t) = E O (x, t) (7)
which includes the previous formulation (6). The observation error E O simultaneously represents the imperfection of À and the measurement errors. E m , E b and E O are assumed to be Gaussian and fully characterized by their covariance matrices Q, B and R.

INRIA

For instance, let X denote a Gaussian stochastic vector depending on a space-time coordinate (x, t); X = X(x, t) and X ′ = X(x ′ , t ′ ) are the values on two given locations. The covariance matrix Σ, computed for X and X ′ , measures their dependency and is defined by: Σ(x, t, x ′ , t ′ ) = (X -X) T (X ′ -X ′ )dP X,X ′ P X,X ′ is the joint distribution of (X, X ′ ) and denotes the expectation.

Variational formulation

In order to solve equations ( 4), ( 5) and ( 7), the following functional, to be minimized, is defined:

E(X) = A A ∂X ∂t + Å(X) T (x, t)Q -1 (x, t, x ′ , t ′ ) ∂X ∂t + Å(X) (x ′ , t ′ )dxdtdx ′ dt ′ + A A À(X, Y) T (x, t)R -1 (x, t, x ′ , t ′ )À(X, Y)(x ′ , t ′ )dxdtdx ′ dt ′ + Ω Ω X(x, 0) -X b (x) T B -1 (x, x ′ ) X(x ′ , 0) -X b (x ′ ) dxdx ′ (8) with A = Ω×[0, T], Ω
and by introducing an auxiliary variable λ, called the adjoint variable in the literature of Data Assimilation:

λ(x, t) = A Q -1 (x, t, x ′ , t ′ ) ∂X ∂t + Å(X) (x ′ , t ′ )dx ′ dt ′ (10)
We detail in the appendix A the determination of the Euler-Lagrange equation associated to (8). This leads to the following system:

λ(x, T) = 0 (11) - ∂λ ∂t + ∂Å ∂X * λ = - A ∂À ∂X * (x, t)R -1 À(X, Y)(x ′ , t ′ )dx ′ dt ′ (12) X(x, 0) = Ω Bλ(x ′ , 0)dx ′ + X b (x) (13) ∂X ∂t + Å(X) = A Qλ(x ′ , t ′ )dx ′ dt ′ (14)
Equation ( 12) makes use of two adjoint operators denoted ∂Å ∂X * and ∂À ∂X * .

These operators are formally defined as the differential operators in the dual RR n 6879 space of X. For a given operator à applied on X, we have:

∂à ∂X (η) T λdxdt = η T ∂à ∂X * (λ)dxdt (15)
for all integrable functions η and λ. If ∂à ∂X is a differential operator, the adjoint operator represents a compact notation for integration by parts. Riesz's theorem ensures the existence and uniqueness of the adjoint operator. For clarifying the discussion, let us determine the adjoint operator of

à = ∂ ∂x in an interval [a, b]: b a ∂f ∂x (x)g(t)dx = [f (x)g(x)] x=b x=a - b a f (x) ∂g ∂x (x)dx = b a f (x) (δ(x -b) -δ(x -a)) g(x) - ∂g ∂x (x) dx = b a f (x) ∂g ∂x * (x)dx
The adjoint operator of

∂ ∂x in [a, b] is then ∂ ∂x * = δ(x -b) -δ(x -a) - ∂ ∂x .

Incremental algorithm

At this stage, a difficulty occurs: the state vector is determined by equations (13,14) using the adjoint variable and the adjoint variable is determined by equations (11,12) using the state vector. To break this deadlock, an incremental method is defined. The underlying idea comes from the following lemma:

min w∈V(w0) E(w) = min δw∈V(0) E(w 0 + δw)
where w 0 denotes a local minimum of E and V(w 0 ) denotes one neighborhood of w 0 . The state vector is therefore written as X b + δX where X b is a mean term, called the background variable in the Data Assimilation literature, and δX is the incremental variable. X is replaced by X b + δX in equations ( 12), ( 13) and ( 14). If Å and À are linear operators, we have:

Å(X) = Å(X b ) + ∂Å ∂X X b (δX) (16) À(X) = À(X b ) + ∂À ∂X X b (δX) (17)
In case of non linearity, they could however be approximated by a first order Taylor development resulting in (16,17). In both cases, this leads to the following

INRIA

new system of equations:

λ(x, T) = 0 (18) - ∂λ ∂t + ∂Å ∂X X b * λ = - A ∂À ∂X X b * R -1 À(X b , Y) + ∂À ∂X X b (δX) dx ′ dt ′ (19) X b (x, 0) = X b (x) (20) ∂X b ∂t + Å(X b ) = 0 (21) δX(x, 0) = Ω Bλ(x ′ , 0)dx ′ (22) ∂δX ∂t + ∂Å ∂X X b (δX) = A Qλ(x ′ , t ′ )dx ′ dt ′ (23)
The background variable X b is first calculated from equations ( 20) and ( 21) independently of the adjoint variable and the incremental variable (direct run).

The adjoint variable λ is then obtained from the background variable using equations ( 18) and ( 19) and the incremental variable δX is obtained from the adjoint variable using equations ( 22) and ( 23). If Å and À are not linear, equations (18,19,22,23) produce an approximated solution X b + δX and an iterative algorithm is used to obtain an exact solution. The iterative algorithm is summarized in the following:

1. Initialization (i = 0) :

(a) Compute the background variable X 0 b (providing the value of state vector at the first iteration):

X 0 b (x, 0) = X b (x) ∂X 0 b ∂t + Å(X 0 b ) = 0, ∀t ∈ [0, T] (b) Initialize the incremental variable: δX(x, t) = 0, ∀t ∈ [0, T] 2. Iteration i > 0 : (a) Compute the adjoint variable λ: λ(x, T ) = 0 - ∂λ ∂t + ∂Å ∂X X i b * (λ) = - ∂À ∂X X i b ,Y * R -1 À(X i b , Y) + ∂À ∂X X i b ,Y (δX) dx ′ dt ′ (b) Compute the incremental variable δX: δX(x, 0) = B(x, x ′ )λ(x ′ , 0)dx ′ ∂δX ∂t + ∂Å ∂X X i b (δX) = Q(x, t, x ′ , t ′ )λ(x ′ , t ′ )dx ′ dt ′ RR n 6879
(c) Update the background value (providing the value of the state vector at iteration i + 1):

X i+1 b (x, t) ⇐ X i b (x, t) + δX(x, t)
and back to step 2 until convergence.

Assimilation of images

This section explains how to solve the ill-posed Image Processing problems described in Section 2 using the framework of Data Assimilation and constitutes the core of this research and the main contribution of the paper. Using Data Assimilation to solve Image Processing Problem is an emerging domain. Studies have been done about curve tracking (Papadakis and Mémin, 2007) and determination of optical flow (Papadakis et al., 2007a,b). In et al [START_REF] Huot | Assimilation of sst satellite images for estimation of ocean circulation velocity[END_REF], a method is proposed to estimate the ocean surface circulation from SST images: assimilating image data in a dynamic image model for producing pseudoobservations for the oceanographic model. In this paper, we consider the general case of ill-posed problems, solved using Tikhonov regularization methods, and we define a method to convert them, in a generic way, from Tikhonov regularization to the Data Assimilation framework. The proposed method rewrites the Image Model and the regularization term under the form of the system of three equations (4,5,7) and we study possible choices for the evolution model and the error's covariance, and their impact in term of spatio-temporal regularization. First, state and observation vectors have to be defined. Obviously, the observations will be images or processed images, but the content of the state vector will strongly depends on the studied problem. For example, segmentation, denoising and restoration use a state vector which is an image. Tracking, image registration and motion estimation use a vector field. Active contours use a curve. Sometimes, the link between the state vector and the observation is highly indirect. For instance, as it is not possible to deduce the ocean circulation from surface temperatures with a shallow-water model, an additional system has to be defined with relevant evolution and observation equations [START_REF] Huot | Assimilation of sst satellite images for estimation of ocean circulation velocity[END_REF].

To solve an Image Processing problem using the 4D-Var algorithm as described in Section 3, a suitable equation describing the temporal evolution of the state vector has to be stated. Next, an observation equation is written expressing the links between the state vector and the images. Last, the initial condition and its background error should be defined. This is discussed in the next Subsections.

The evolution model

The evolution equation describes the temporal dynamics of the state vector and can be used, for instance, to impose a temporal regularity for recovering result in case of missing data. A first and simple possibility to specify it could be to assume X being constant over time, which can be expressed by:

dX dt = 0

INRIA

This equation is developed using the chain rule:

∂X ∂t + ∂X ∂x ∂x ∂t = 0 (24)
which is a transport equation. ∂x ∂t is a velocity vector and the evolution model

is Å(X)(x, t) = ∂X ∂x ∂x ∂t
. An example of using ( 24) as evolution equation is given in Section 5 to determine optical flow.

Another possible specification is to express the transport of the state vector as a diffusion process, a physical law ruling the transport of chemical species or temperature. The general formulation is: ∂X ∂t = ∇ T (D∇X) and by identification the evolution model is: Å(X) = -∇ T (D∇X). The matrix D is a tensor characterizing simultaneously the direction and the intensity of the diffusion. If D does not depend on spatial coordinates, the diffusion is linear and equivalent to a smoothing process using a Gaussian convolution [START_REF] Witkin | Scale-space filtering[END_REF]. It is also possible to drive the diffusion according to image characteristics. A standard example is the Perona & Malik diffusion [START_REF] Perona | Space scale and edge detection using anisotropic diffusion[END_REF]): the tensor matrix D is equal to c( ∇X )Id with c a Gaussian function and Id the identity matrix with the result of smoothing the image on homogeneous regions and preserving contours. The tensor matrix can be chosen by taking into account the orientation of the image gradient [START_REF] Weickert | Anisotropic diffusion in image processing[END_REF][START_REF] Weickert | Applications of nonlinear diffusion in image processing and computer vision[END_REF] resulting in spatial regularization properties similar to that of Nagel described in Section 2.

The two previous definitions of Å remain restrictive for image sequences displaying complex temporal dynamics. In this case, the evolution model could be built using prior information on the scene. For example, dynamics can be approximated by piecewise linear functions whose parameters are estimated by analyzing the evolution of image probability density function with a particule filter method. However, this issue remains complex and widely open.

An evolution model based on a physical law can also be considered. In [START_REF] Huot | Assimilation of sst satellite images for estimation of ocean circulation velocity[END_REF][START_REF] Papadakis | Image assimilation for motion estimation of atmospheric layers with shallow-water model[END_REF] a shallow-water equation is chosen for modeling the temporal evolution of apparent motion. This is an advection-diffusion equation with additional forcing terms. However, this is specific to the ocean surface circulation and can not be applied to a generic framework. In particular, it is unsuitable to the application described in Section 5.

In addition to the evolution model, the matrix Q of equation ( 14) has to be specified. It is first important to understand the action of Q -1 in the functional (8). Let us consider three possible choices for Q and analyze their respective impact in a functional F T (X)Q -1 F (X)dxdx ′ which has to be minimized. As a first example, let Q be the Dirac covariance defined by Q(x, x ′ ) = δ (xx ′ ). The Dirac function, δ (), has the following property:

Ω δ (x ′ ) δ (x -x ′ ) dx ′ = δ (x)
The inverse of a covariance matrix is formally and implicitly defined [START_REF] Oliver | Calculation of the inverse of the covariance[END_REF] by:

Σ -1 (x, x ′′ )Σ(x ′′ , x ′ )dx ′′ = δ(x -x ′ ) (25)
By identification, we have Q -1 (x, x ′ ) = δ (xx ′ ) and:

Ω 2 F (X) T (x)Q -1 (x, x ′ )F (X)(x ′ )dxdx ′ (26) = Ω F (X) T (x)F (X)(x)dx = Ω F (X) 2 dx
A Dirac covariance is therefore acting like a zero-order regularization.

More generally, we consider the case of an isotropic covariance which is written as Q(x, x ′ ) = q(xx ′ ). In this case, determining its inverse from ( 25) is equivalent to determine its inverse convolution defining by:

q -1 ⋆ q(x) = δ(x) (27)
This is more easily done in the Fourier domain as the latter equation is equivalent to:

q -1 ⋆ q(ω) = 1
Using the convolution theorem, we have:

q -1 (ω) q(ω) = 1 q -1 (ω) = 1 q(ω)
and the inverse convolution is given by the inverse Fourier transform of 1 q .

Let us apply this with the exponential covariance defined by q

(x) = exp - |x ′ | σ .
Its Fourier transform is 2σ 1 + σ 2 ω 2 . We have:

q -1 (ω) = 1 + σ 2 ω 2 2σ q -1 (x) = 1 2σ δ (x) -σ 2 δ ′′ (x)
We replace the expression of Q -1 (x, x ′ ) by q -1 (xx ′ ) in the functional (26):

Ω 2 F (X) T (x)Q -1 (x, x ′ )F (X)(x ′ )dxdx ′ = 1 2σ Ω F (X) T (x) F (X) -σ 2 ∂ 2 F (X) ∂x 2 dx = 1 2σ Ω F (X) 2 + σ 2 ∂F (X) ∂x 2 dx (28)
Line ( 28) is obtained from the previous using integration by parts. The exponential covariance is associated to a first-order regularization.

A more complex possibility is the Gaussian covariance defined by q(x) = exp -x 2 σ 2 . Using the previous technique, the Fourier transform of the Gaussian function is first established:

q(ω) = σ √ π exp(- σ 2 4 ω 2
). The Fourier transform of the inverse covariance is thus q -1 (ω) = √ π σ exp σ 2 4 ω 2 but its inverse

INRIA

Fourier transform can not be directly computed. Let us replace the exponential by its infinite series:

q -1 (ω) = √ π σ n≥0 1 n! σ 2 4 ω 2 n = √ π σ n≥0 a n ω 2n (29) with a n = 1 n! σ 4 2n .
Remembering that 1 is the inverse Fourier transform of the Dirac function and using the theorem of Fourier derivation, equation ( 29) leads to:

q -1 (x) = √ π σ n≥0 a n (i) 2n ∂ 2n ∂x 2n δ(x) = √ π σ n≥0 a n (-1) n δ (2n) (x)
Let us examine the impact of such covariance in the functional (26):

F (X) T (x)Q -1 (x, x ′ )F (X)(x ′ )dxdx ′ = n≥0 (-1) n a n F (X) T (x) δ (2n) (x -x ′ )F (X)(x ′ )dx ′ dx = n≥0 (-1) n a n F (X) T (x)F (2n) (X)(x)dx
For the term of order n, integration by parts is applied n times:

(-1) n F (X) T (x)F (2n) (X)(x)dx = F (n) (X) T F (n) (X)dx
We finally obtain:

F (X) T (x)Q -1 (x, x ′ )F (X)(x ′ )dxdx ′ = n≥0 a n ∂ n F (X) ∂x n 2 (x)dx
and the Gaussian exponential covariance corresponds to the Tikhonov regularization at any order. The inversion of a covariance matrix is however non-trivial and usually inaccessible. Restrictive choices have to be made such as those previously described. In the general case, it still remains possible to approximate the matrix of covariance using finite difference operators and to inverse the matrix using numerical techniques. Unfortunately, if the discrete matrix is large, its inversion is costly and often numerically unstable. For further details, the reader is referred to [START_REF] Oliver | Calculation of the inverse of the covariance[END_REF][START_REF] Tarantola | Inverse Problem Theory and Methods for Model Parameter Estimation[END_REF].

In conclusion, the type of regularization applied on the state vector -a zeroorder, a first-order or any order -has to be chosen by selecting the suitable matrix of covariance (Dirac, exponential or Gaussian). A spatial regularization on X can not be included in the model error as Q describes the error on the evolution equation and not on the state vector itself. However, this regularization on X can be included in the observation error if required.

The observation equation

As previously pointed out, the observation equation describes the links between the state vector and the observation. In the standard framework of Image Processing, an image or a sequence of images provides the observation in the Image Model Á expressed in equation (1). The operator À, as it appears in equation ( 7) (to be more general than ( 6)), should then be directly defined as the image operator, i.e. À ≡ Á.

The observation error, quantified by its matrix of covariance R, has also to be specified. This is a possible mean for not taking into account missing data: as R weights the observation influence in equation ( 19), its inverse should have values close to zero when observations should be discarded. Using the Dirac covariance (which expresses a null interaction between two space-time locations), R is then written as: R(x, t,

x ′ , t ′ ) = r(x, t)δ (x -x ′ ) δ (t -t ′ )
with r a real matrix. The inverse is:

R -1 (x, t, x ′ , t ′ ) = δ (x -x ′ ) δ (t -t ′ ) r -1 (x, t) (30)
The function r -1 characterizes the quality of the observation: a high value indicates that the observation value is relevant and a value close to zero indicates an irrelevant observation value, which should not included in the computation of the solution. Assuming the availability of a function f measuring the confidence in observation values (f ∈ [0, 1], f = 0 for no confidence), one possible formulation of r -1 is:

r -1 (x, t) = r 0 (1 -f (x, t)) + r 1 f (x, t) (31)
r -1 (x, t) will be equal to a "minimal value" r 0 where confidence is 0 and be equal to a "maximal value" r 1 where confidence is 1. Matrices r 0 and r 1 are chosen to be constant and invertible. For completeness, f is modeled as:

f (x, t) = f sensor (x, t)f noise (x, t)f À (x, t) (32)
f sensor indicates the availability of observation values: it is equal to 0 when data are not or wrongly acquired by the sensor. f noise characterizes data quality: it is close to 0 for noisy data. f À measures the confidence in the observation model;

it is close to 0 if the observation equation is not valid. Observation values with a low confidence will then not be considered for the computation of the state vector and solution of the Image Processing problem.

Conclusion

In this Section, we considered the class of ill-posed Image Processing problems which are usually solved by constraining spatially and temporally the solution. We proposed a generic method to convert the Image Model and the spatiotemporal regularity constraint in term of Data Assimilation components. The Image Model is taken as the observation model and the observation error weights the importance of the observation values in the computation of the state vector: when data are irrelevant, their values are no more used. Of course, this is only possible if the quality of the observation can be evaluated. The evolution INRIA equation describes the temporal evolution of the state vector and we proposed two generic choices for the operator Å, depending on the application domain.

However, both are too restrictive to deal with complex temporal dynamics. The spatial regularity of the state vector is ensured by the covariance matrix Q and we gave three examples of generic regularization. In the next Section, we show how to use Data Assimilation to determine optical flow by applying these general principles.

Application to optical flow estimation

Let I be a sequence of images on a bounded domain of Ê 2 , denoted Ω. Let W(x, t) be the velocity vector of a point x ∈ Ω between frame t and t + △t verifying:

I(x + W(x, t)△t, t + △t) = I(x, t) ∀x ∈ Ω (33)
As this equation is non linear with respect to W, the left member of equation ( 33) is often linearized using a first order Taylor development around △t = 0. This provides the so-called optical flow constraint equation [START_REF] Horn | Determining optical flow[END_REF]:

∇I T (x, t)W(x, t) + ∂I ∂t (x, t) = 0 ∀x ∈ Ω (34)
Estimating apparent motion is an ill-posed problem: the velocity vector has two components and the optical flow equation is not sufficient to compute both. A solution could be obtained in the Image Processing context using a Tikhonov regularization as stated in Section 2 by constraining spatially [START_REF] Horn | Determining optical flow[END_REF] or spatially and temporally [START_REF] Weickert | Variational optic flow computation with a spatio-temporal smoothness constraint[END_REF] the solution. Equation ( 34) is often preferred to (33) because it is linear and the associated Euler-Lagrange equation can be discretized using an explicit numerical scheme. However, it is possible to directly use the equation (33) to determine the optical flow. This has been described in [START_REF] Brox | High accuracy optical flow estimation based on a theory for warping[END_REF] where the resulting Euler-Lagrange equation has been approximated using a semi-implicit scheme. The advantage of equation ( 33) is that W is correctly estimated even for high velocity values. Equation ( 34) is an approximation and only well suited for low velocity except if incremental algorithms [START_REF] Odobez | Direct incremental model-based image motion segmentation for video analysis[END_REF][START_REF] Proesmans | Determination of optical flow and its discontinuities using non-linear diffusion[END_REF] or space-scale methods [START_REF] Alvarez | Reliable estimation of dense optical flow fields with large displacements[END_REF] are considered.

In this paper, we choose to consider the optical flow constraint (34) in order to describe the application of the Data Assimilation framework to solve a standard Image Processing problem and to compare it with state-of-the-art methods.

Observation and evolution models

The optical flow constraint is used as the observation equation. With this choice of observation model, the field W(x, t) of velocity vectors is now considered as the state vector X(x, t) and the sequence of image gradients (∇I(x, t), I t (x, t)) constitutes the observation vector Y(x, t). By identification of ( 7) and ( 34), the observation model is:

À(W, I)(x, t) = ∇I(x, t) T W(x, t) + I t (x, t) (35)
We have to define an appropriate observation error. Equations ( 30) and ( 31) are used to define the inverse of R and to locate the observation values which must not be assimilated. The observation model being scalar, the matrices r 0 and r 1 are scalars and respectively set to ǫ and 1ǫ with ǫ ≃ 10 -6 . Equation ( 32) is used as the observation confidence. Without any information, f noise is assumed to be equal to 1. f sensor is set to 0 where data are not acquired and to 1 otherwise. f À is chosen from the following remark: the spatio-temporal gradient is null on regions of uniform grey level values and equation ( 34) is then degenerated. For avoiding considering these points, f À is defined by:

f À (x, t) = 1 -exp(-∇ 3 I(x, t) 2 ) (36)
where ∇ 3 denotes the spatio-temporal gradient operator.

The transport of the velocity, equation ( 24), is chosen as the evolution equation:

∂W ∂t + ∇W T W = 0 ( 37
)
This equation is rewritten as the two-component system:

∂U ∂t + U U x + V U y = 0 (38) ∂V ∂t + U V x + V V y = 0 (39)
and the evolution model is:

Å(W) = Å 1 (W) Å 2 (W) = U U x + V U y U V x + V V y
The Q matrix is chosen as:

Q(x, t, x ′ , t ′ ) = exp - 1 σ ( x -x ′ + |t -t ′ |) 1 0 0 1 (40)
The initial condition of the state vector has also to be provided. We make use of Horn and Schunck's algorithm [START_REF] Horn | Determining optical flow[END_REF] to compute the velocity field on the two first frames of the sequence. We assume the initial condition to be without errors and we set B(x, x ′ ) = δ(x-x ′ ) as the background error.

Adjoint operators

In order to determine the adjoint operators for Å and À, the directional derivatives must first be established. Using the definition (9), we obtain:

∂Å 1 ∂W (η) = ∂Å 1 ∂U (η 1 ) + ∂Å 1 ∂V (η 2 ) = U η 1 x + U x η 1 + V η 1 y + U y η 2 ∂Å 2 ∂W (η) = ∂Å 2 ∂U (η 1 ) + ∂Å 2 ∂V (η 2 ) = U η 2 x + V y η 2 + V η 2 y + V x η 1
with η = η 1 η 2 T and η x and η y standing for partial derivate with respect to x and y. The reader is referred to appendix B.1 for more details. Using (15),

INRIA

integration by parts and considering border terms to be equal to zero, the adjoint operator of Å is:

∂Å 1 ∂W * (λ) = -U λ 1 x -V y λ 1 -V λ 1 y + U y λ 2 ∂Å 2 ∂W * (λ) = -U x λ 2 -U λ 2 x -V λ 2 y + V x λ 1 with λ = λ 1 λ 2 T . Details are reported in appendix B.2.
The directional derivative of the observation operator is:

∂À ∂W (η)(x, t) = ∇I T (x, t)η(x, t)
and determining the adjoint operator is direct, as described in appendix B.3:

∂À ∂W * (λ)(x, t) = ∇I(x, t)λ(x, t)

Discretization

Using the choices made in Subsection 5.1, the three PDEs (19,21,23) become:

∂W ∂t + W T ∇W = 0 (41) - ∂λ ∂t -∇λ T W -λ T ∇ ⊥ W = - 1 2σ ∇IR -1 ⋆ L (42) ∂δW ∂t + W T ∇δW + ∇W T δW = Q ⋆ λ (43) with ∇ ⊥ W = ∇ ⊥ U ∇ ⊥ V , ∇ ⊥ U = U y -U x T and L = I t + ∇I T (W + δW).
The covariance matrices Q and R being chosen isotropic, they depend on coordinates (xx ′ , tt ′ ), and the right members of equations ( 42) and ( 43) can then be expressed as a convolution product. These three equations are discretized using the finite difference technique. Let us first examine equation ( 41): it is a 2D non linear advection equation. The advection term corresponds to the velocity transport. Its direct approximation by a standard Euler scheme is numerically unstable. This instability has several origins: the non-linearity of the equation, the multi-dimensionality of the state vector, the simultaneous occurrence of linear and non linear terms in the equation. For stabilizing the numerical scheme, a diffusive term may be introduced into the equation:

∂W ∂t + ∇W T W = κ∇ 2 W
with κ > 0 having a small value as done in [START_REF] Papadakis | Image assimilation for motion estimation of atmospheric layers with shallow-water model[END_REF]. This is known under the name of Lax-Wendroff method. Such an equation can be approximated using an explicit Euler scheme (with Courant-Friedrich-Levy condition) or implicit Euler scheme. This has the drawback to smooth the solution of the equation and not preserving discontinuities. We propose, in the following, a stable numerical scheme for the advection equation without adding a diffusive term. As W(x, t) is a vector of Ê 2 , equation ( 41) has two components. The first component combines a term of linear advection in direction y and non linear advection in direction x. It can be expressed as a two-equation system using a splitting method [START_REF] Verwer | A note on operator splitting in a stiff linear case[END_REF]:

∂U ∂t + U U x = 0 (44) ∂U ∂t + V U y = 0 (45)
Equation ( 44) is rewritten with the Lax-Friedrich method [START_REF] Sethian | Level Set Methods[END_REF] as

∂U ∂t + ∂F (U ) ∂x = 0 with F (U ) = 1 2 U 2
. This new equation is discretized by the following explicit scheme:

U k+1 i,j = 1 2 (U k i+1,j + U k i-1,j ) - △t 2 (F k i+1,j -F k i-1,j ) with U k i,j = U (x i , y i , t k ), F k i,j = F (U (x i , y i , t k ))
and △t the time step. The term 1 2 (U k i+1,j + U k i-1,j ) stabilizes the scheme (by adding a diffusive effect) when △t has a low value (the Courant-Friedrich-Levy condition). The linear advection ( 45) is discretized using an explicit shock scheme [START_REF] Sethian | Level Set Methods[END_REF]:

U k+1 i,j = U k i,j -△t max(V k i,j , 0) U k i,j -U k i-1,j + min(V k i,j , 0) U k i+1,j -U k i,j
It can be seen that the second component of ( 41) contains a linear advection term in direction x and a non linear one in direction y. The same strategy is then applied for discretization. Equation ( 42) combines a linear advection, a term of reaction and a forcing term. It also has two components. The first is -

∂λ 1 ∂t -U λ 1 x -V y λ 1 -V λ 1 y + U y λ 2 = ∂I ∂x A with A = -1 2σ R -1 ⋆ L.
It is split into two parts. The first part contains the linear advection in direction x and the reaction term -∂λ 1 ∂t -

U λ 1

x -V y λ 1 = 0 and is discretized in the same way as (45) with an explicit shock scheme. However, the equation is retrograde as the initial condition is given at time T :

(λ 1 ) k-1 i,j = 1 + 1 2 (V k i,j+1 -V k i,j-1 ) (λ 1 ) k i,j + △t max(U k i,j , 0)((λ 1 ) k i,j -(λ 1 ) k i-1,j ) + min(U k i,j , 0)((λ 1 ) k i+1,j -(λ 1 ) k i,j
)) The second part contains the linear advection term in direction y and the forcing term: -

∂λ 1 ∂t -V λ 1 y + U y λ 2 = ∂I ∂x A.
Again, an explicit shock scheme is used:

(λ 1 ) k-1 i,j = (λ 1 ) k i,j - △t 2 U k i,j+1 -U k i,j-1 (λ 2 ) k i,j -△t(I x B) k i,j - △t max(V k i,j , 0)((λ 1 ) k i,j -(λ 1 ) k i,j-1 ) + min(V k i,j , 0)((λ 1 ) k i,j+1 -(λ 1 ) k i,j
) Having the same structure, the second component of ( 42) is discretized with the same method. The complete numerical scheme is described in Appendix C.2.

The last equation, (43), is similar to equation ( 42): a linear advection with a reaction term and a forcing term. We therefore use the same discretization technique. The full numerical scheme is shown in Appendix C.3.

INRIA

Results

The "taxi" sequence and a synthetic sequence have been chosen for discussing results. For both cases, image gradients are computed with a convolution method and a derivative Gaussian kernel, the parameter σ of the matrix Q -Eq (40)and the Gaussian kernel variance are set to 1, the number of iterations is set to 5.

The taxi sequence displays several cars moving with a slow and quasi uniform motion.

In a first experiment we compute the optical flow using the Data Assimilation method with image gradients as observations. Horn & Schunk's method is also applied on the sequence and both results are displayed for comparison purposes. Figures 1, 2 and3 show the results obtained on three frames of the sequence. Results are very similar illustrating that both methods are equivalent when observation values are available on the whole sequence. Having chosen Horn & Schunck or another method does not really matter. The paper's issue is not to compare optical flow methods but to prove the efficiently of Data Assimilation for dealing with missing data and complex dynamics. Then, a second experiment is designed in case of missing data. A large region around the white car, denoted ℜ, is set to zero on one frame of the sequence to simulate a sensor failure. To indicate to the algorithm the irrelevance of pixel values inside this region, the function f sensor returns the value 0 inside ℜ and the value 1 outside. The spatiotemporal gradient is then computed on the modified sequence and provides the observations. Figure 4 shows the results with Data Assimilation and Horn & Schunk methods. This latter obviously fails to provide acceptable velocity vectors over ℜ while Data Assimilation provides a correct result thanks to the eviction of missing observation in the computation. A similar experimentation is performed using several small regions set to zero on one frame and flagged as not acquired by f sensor . This lack of observation also disturbs Horn & Schunk's algorithm while Data Assimilation provides a correct result as illustrated by Figure 5. Even a whole frame of the observation sequence can be missing: we force image gradients to zero on the fifth frame of the taxi sequence resulting to f À = 0 on this frame. In this case, Horn & Schunk's method can not provide any result on this frame. Figure 6 is then comparing results obtained by Data Assimilation with and without observation on frame 5. Results remain similar, due to the fact that the evolution model correctly describes the temporal dynamics of the sequence and compensates the missing data.

A third experiment is dedicated to the demonstration that the evolution model allows a better use of the dynamics in the estimation process. For that purpose, we built a synthetic sequence showing one square moving horizontally and one moving vertically. At the end of the sequence, the two squares meet each other.

Conclusion

In this paper we proposed a general framework to solve ill-posed Image Processing problems by Data Assimilation methods. This approach is an alternative to Weickert's method, which constrains the solution's variations in space and time.

In the case where the sequence dynamics are approximately known and can expressed as an evolution equation, we explained in the paper how this information can be used, simultaneously with the observation equation, in the framework of Data Assimilation to temporally constrain the solution and obtain a better result. Moreover the knowledge of the temporal dynamics has also being proven useful to handle the problem of missing and noisy data. For that, the observation equation is associated to a specific observation covariance matrix, with high values on pixels of missing and noisy data, which are then not considered during the computation process. On these pixels, the solution is mainly obtained by the evolution equation. Obviously, this is only possible if a confidence measure on observation data is available. The 4D-var algorithm described in the paper is a frame-by-frame process and has the advantage of low memory requirements compared to a Tikhonov spatiotemporal regularization method, which requires to include the whole sequence in the computation.

The three previous concerns have been illustrated on the issue of estimating the optical flow on a sequence of images by assimilating image gradient observations within a model describing the evolution of velocity. In this application, the evolution equation is the transport of velocity by itself. This is simple and correctly describes the dynamics, which are locally translational and uniform. As this is a non linear equation, the capability of Data Assimilation to solve non linear models is fully exploited. The chosen observation equation used in this application is the optical flow constraint equation, linearly linking velocity and image brightness gradients.

The discretization of the velocity transport equation leads to an unstable numerical scheme when standard techniques of finite differences are used. To overcome this problem, a robust numerical scheme, based on a splitting methods, has been proposed in the paper.

A first perspective of this work is to investigate other evolution equations. For some applications it is possible to use an evolution equation which is physically justified: for instance, ocean surface motion is described with a shallowwater model. If we are concerned with video sequences of rigid objects, the transport of velocity by itself is relevant for locally translational displacements, but becomes inadequate for rotational or discontinuous displacements. A possible solution is to build a parametric model of the image dynamics: the image sequence's spatio-temporal domain is divided in several domains, each one describing a local evolution equation approximating correctly the image dynamics. Then, the transport of velocity by itself can be used for each domain but with specific initial conditions. A second perspective is to investigate the spatial regularization of state vector X through the observation covariance matrix R as explained in Subsection 4.1. A last perspective addresses the experimental application chosen in the paper: the estimation of optical flow. The optical flow constraint equation has been used as observation equation. This equation has the advantage to be linear but it is only an approximation of the transport of image brightness by velocity. The initial transport equation, even if non linear, can be used as observation equation: the differential and adjoint operator of the observation model can be determined if the image brightness (the observation vector) is supposed to be differentiable. The 4D-var algorithm described in Subsection 3.3 and designed to deal with non linear observation and evolution models, could then be used with such observation equation.

A The Euler-Lagrange equation of E (8)

Let us first compute the derivative of E in direction η using definition (9):

E(X + γη) = A A (X t + γη t + Å(X + γη)) T Q -1 (X t + γη t + Å(X + γη)) T dxdtdx ′ dt ′ + A A À(X + γη) T R -1 À(X + γη)dxdtdx ′ dt ′ + Ω Ω (X + γη -X b ) T B -1 (X + γη -X b )dxdx ′ d dγ E(X + γη) = A A η t + d dγ Å(X + γη) T Q -1 (X t + γη t + Å(X + γη)) dxdtdx ′ dt ′ + A A d dγ À(X + γη) T R -1 À(X + γη)dxdtdx ′ dt ′ + Ω Ω γ T B -1 (X + γη -X b )dxdx ′

INRIA

Let us γ tend to zero:

∂E ∂X (η) = A A η t + ∂Å ∂X (η) T Q -1 (X t + Å(X)) dxdtdx ′ dt ′ + A A ∂À ∂X (η) T R -1 À(X)dxdtdx ′ dt ′ + Ω Ω η T B -1 (X -X b )dxdx ′
We use integration by parts in order to factorize each term with η T :

∂E ∂X (η) = A A η T δ(t = T) -δ(t = 0) - ∂ ∂t + ∂Å ∂X * Q -1 (X t + Å(X)) dxdtdx ′ dt ′ + A A η T ∂À ∂X * R -1 À(X)dxdtdx ′ dt ′ + Ω Ω η T B -1 (X -X b )dxdx ′
Let us introduce the adjoint variable λ in the previous expression and use Fubbini's theorem:

∂Å ∂X (η) = Ω η T (x, T)λ(x, T)dx - Ω η T (x, 0)λ(x, 0)dx + A η T (x, t) - ∂λ ∂t + ∂Å ∂X * (λ) dxdt + A η T (x, t) A ∂À ∂X * R -1 À(X)dx ′ dt ′ dxdt + Ω η T (x, 0) Ω B -1 (X(x ′ , 0) -X b (x ′ ))dx ′ dx
A solution of ∂Å ∂X (η) = 0 ∀η w.r.t. X verifies the following system:

λ(x, T) = 0 (46) -λ(x, 0) + Ω B -1 (X(x ′ , 0) -X b (x ′ ))dx ′ = 0 (47) - ∂λ ∂t + ∂Å ∂X * (λ) + A ∂À ∂X * R -1 À(X)dx ′ dt ′ = 0 (48)
Using the definition of inverse covariance (25), Equation ( 47) is rewritten as:

X(x, 0) = X b (x) + Ω B(x, x ′ )λ(x ′ , 0)dx (49)
and the state vector is expressed from λ using (10):

∂X ∂t

+ Å(X) = A Q(x, t, x ′ , t ′ )λ(x ′ , t ′ )dx ′ dt ′ (50)
Equations ( 46), ( 48), ( 49) and ( 50) provide the Euler-Lagrange equations. The operator Å is defined by: Å

(W) = W T ∇W = Å 1 (W) Å 2 (W) = U U x + V U y U V x + V V y
The differential of Å is formally equal to the following Jacobian matrix:

∂Å ∂W =     ∂Å 1 ∂U ∂Å 1 ∂V ∂Å 2 ∂U ∂Å 2 ∂V     (51)
Each element of this matrix is obtained using the definition of the directional derivative. Considering the first element of ∂Å ∂W , the directional derivative with respect to U in direction η 1 is given by:

∂Å 1 ∂U T η 1 = lim γ→0 d dγ Å 1 (U + γη 1 , V )
and we obtain:

∂Å 1 ∂U T η 1 = U η 1 x + V η 1 y + U x η 1 ∂Å 1 ∂V T η 2 = U y η 2 ∂Å 2 ∂U T η 1 = V x η 1 ∂Å 2 ∂V T η 2 = U η 2 x + V η 2 y + V y η 2 ∂Å ∂W = U ∂ x + V ∂ y + U x U y V x U ∂ x + V ∂ y + V y
In equation ( 23), the differential of Å appears as a directional derivative in direction δW and it has been defined in such a way (see equation ( 51)) that the Jacobian matrix does not need to be transposed to compute the derivative in a given direction.

+ V x δU + V y δV T λ 1 λ 2 dx = -δU ∂ x (U λ 1 ) -δU ∂ y (V λ 1 ) + δU U x λ 1 + δV U y λ 1 δU V x λ 2 -δV ∂ x (U λ 2 ) -δV ∂ y (V λ 2 ) + δV V y λ 2 dx = δU δV -U λ 1 x -V y λ 1 -V λ 1 y + V x λ 2 -U x λ 2 -U λ 2 x -V λ 2 y + U y λ 1 dx = δU δV -U ∂ x -V ∂ y -V y V x U y -U ∂ x -V ∂ y -V U x λ 1 λ 2 dx
The adjoint operator of ∂Å ∂W is:

∂Å ∂W * = -U ∂ x -V ∂ y -V y V x U y -U ∂ x -V ∂ y -U x
and its expression in direction λ is:

∂Å ∂W * (λ) = -U λ 1 x -V λ 1 y -V y λ 1 + V x λ 2 U y λ 1 -U λ 2 x -V λ 2 y -U x λ 2

B.3 Differential of À and adjoint operator

The operator À is defined by: À(W, I)(x, t) = ∇I(x, t) T W(x, t) + I t (x, t) The differential of À is:

∂À ∂W = ∂À ∂U ∂À ∂V T
The first component is given by: ∂À ∂U η 1 = lim

RR n 6879

The same calculus leads for the second component to ∂À ∂V η 2 = I y η 2 and finally the differential of À is:

∂À ∂W = ∇I T
The adjoint operator is defined by: ∂À ∂W (δW), λ Equation ( 52) is rewritten and split into the system:

F = 1 2 U 2 (54) ∂U ∂t + F x = 0 (55) ∂U ∂t + V U y = 0 (56)
Equations ( 55) and ( 56) are now linear advection equations. They can be approximated using the following schemes:

U k+1 i,j -1 2 (U k i+1,j + U k i-1,j ) △t = - 1 2 (F k i+1,j -F k i-1,j ) U k+1 i,j -U k i,j △t = -S y (V, U ) k i,j
S is the discrete operator approximating the advection operator using a shock filter, defined in the following in the y direction:

S y (V, U ) i,j = max(V i,j , 0) (U i,j -U i,j-1 ) + min(V i,j , 0) (U i,j+1 -U i,j )

INRIA Equation ( 53) is rewritten and split into the system:

G = 1 2 V 2 ∂V ∂t + G y = 0 ∂V ∂t + U V x = 0
and then approximated by:

V k+1 i,j -1 2 (V k i+1,j + V k i-1,j ) △t = - 1 2 (G k i,j+1 -G k i,j-1 ) V k+1 i,j -V k i,j △t = -S x (U, V ) k i,j

C.2 Evolution equation of the adjoint variable

Equation ( 42) has two components and can be rewritten as follow:

λ 1 t -U λ 1 x -V y λ 1 -V λ 1 y + V x λ 2 = I x B (57)
λ 2 t -U λ 2 x -U x λ 2 -V λ 2 y + U y λ 1 = I y B (58)
with B = -1 2σ R -1 ⋆ L. Equation ( 57) is split into:

λ 1 t = U λ 1 x + V y λ 1 λ 1 t = V λ 1 y -V x λ 2 + (I x B)
The numerical scheme is retrograde because the initial condition for λ 1 is given at time t = T. We use an explicit Euler scheme: (λ 1 ) k i,j -(λ 1 ) k-1 i,j △t = S x (U, λ 1 ) k i,j + 1 2 (V k i,j+1 -V k i,j-1 )(λ 1 ) k i,j (λ 1 ) k i,j -(λ 1 ) k-1 i,j △t = S y (V, (λ 1 ) k ) i,j -1 2 (V k i+1,j -V k i-1,j )(λ 2 ) k i,j + (I x B) k i,j

The numerical scheme is written:

(λ 1 ) k-1 i,j = (λ 1 ) k i,j -△t S x (U, λ 1 )) k i,j + 1 2 (V k i,j+1 -V k i,j-1 )(λ 1 ) k i,j
(λ 1 ) k-1 i,j = (λ 1 ) k i,j -△t S y (V, λ 1 ) k i,j -

1 2 (V k i+1,j -V k i-1,j )(λ 2 ) k i,j + (I x B) k i,j
The equation ( 58) is split into:

λ 2 t = U λ 2 x + U x λ 2 λ 2 t = V λ 2
y -U y λ 1 + (I y B) and the numerical scheme is:

(λ 2 ) k-1 i,j = (λ 2 ) k i,j -△t S x (U, (λ 2 )) k i,j + 1 2 (U k i+1,j -U k i-1,j )(λ 2 ) k i,j
(λ 2 ) k-1 i,j = (λ 2 ) k i,j -△t S y (V, λ 1 ) k i,j -1 2 (U k i,j+1 -U k i,j-1 )(λ 1 ) k i,j + (I y B) k i,j

RR n 6879

  Figures 7(b), 8(b) and 9(b) show the result with Horn & Schunk's method which fails to estimate a correct velocity direction when the squares meet. This is due to an over-regularization. Figures 7(a), 8(a) and 9(a) show the result with Data Assimilation: directions are better estimated because the evolution model correctly describes the dynamics and avoids the effects of spatial regularization.

Figure 1 :

 1 Figure 1: Comparison Data Assimilation (left) / Horn-Schunk (right) -frame 3.

Figure 2 :

 2 Figure 2: Comparison Data Assimilation (left) / Horn-Schunk (right) -frame 6.

Figure 3 :

 3 Figure 3: Comparison Data Assimilation (left) / Horn-Schunk (right) -frame 9.

Figure 5 :

 5 Figure 5: Missing data on small regions in frame 5.

Figure 6 :

 6 Figure 6: Missing data on the whole frame 5.

Figure 7 :

 7 Figure 7: Results on synthetic sequence -frame 4.

Figure 8 :

 8 Figure 8: Results on synthetic sequence -frame 7.

  x (U + γη 1 ) + I y V + I t = I x η 1

  being the spatial domain and [0, T] the temporal domain. If E m , E b and E O are assumed to be independent, the functional E represents the log-likelihood of X. The minimization is carried out by solving the associated

	Euler-Lagrange equation. The differential derivative of E with respect to X in direction η: ∂E ∂X	is obtained by computing the
	∂E ∂X	(η) = lim γ→0	d dγ	(E(X + γη))

  It is therefore evaluated as: + V δU y + U x δU + U y δV U δV x + V δV y + V x δU + V y δV is a column vector and has to be transposed to perform the scalar product with λ = λ 1 λ 2 T : U δU x + V δU y + U x δU + U y δV U δV x + V δV y

	B.2 Adjoint operator of	∂Å ∂W
	The adjoint operator of	∂Å ∂W	is formally defined by:
	∂Å ∂W	(δW), λ	L 2	=	δW,	∂Å ∂W	*	(λ)	L 2
						=		∂Å ∂W	(δW)	T	λdx
	The directional derivative (δW) ∂Å ∂Å ∂W ∂W (δW)
	∂Å ∂W	(δW) =	∂Å ∂W	δW =	   	∂Å 1 ∂U ∂Å 2 ∂U	δU + δU +	∂Å 1 ∂V ∂V ∂Å 2	δV δV	   
				=	U δU x INRIA

T λdx =

RR n 6879

Béréziat & Herlin

C.3 Evolution equation of the incremental variable

Equation ( 43) has two components which are expressed as follow:

Equation ( 59) is split into:

Again, linear advection terms are approximated using chock filter.

Equation ( 60) is split into:

and approximated by: