Sophie Le Bellu
email: sophie.lebellu@ensc.fr

Benoit Le Blanc
email: benoit.leblanc@ensc.fr

Main Descriptors of Professional Gestures, Regarding to their Intergenerational Transfer 1

Keywords: Tacit knowledge, video capture, professional gesture, Multimedia Apprenticeship Platform (MAP)

Operational knowledge in industries is difficult to capture because of the tacit nature of gestures. The operators can be equipped with video capture devices. They can also be invited to describe the intentions of their gesture. But the bottleneck is in the gesture itself. This paper describes an experiment, which was performed at EDF (Electricité de France). Several categories of professional gestures were studied, and 10 of them were investigated. For the capture phase, the protocol aims to approach the intentions of operators, their mental models. We have produced training video material called "MAP" (Multimedia Apprenticeship Platform) for novices to learn and internalize the practice. An anthropocentric three-step methodology was applied. For all the gestures, eight descriptors were used to qualify the environment and the situation: realism, reproducibility, length, granularity, mobility, interaction, visual salience and haptic salience. A four-value scale is used on each descriptor to quantify the criticism of the gestures. Results include the perimeter of application of the MAP methodology, describing professional gestures which can be captured, formalized and transmitted. The majority of the descriptors have a moderated influence on the process, concentrating their impact on one point of the method. Only two descriptors have high-level impact on the whole process: interaction between operators and visual dimension during the realization of the professional gesture.

Introduction, context

This present work takes place in a knowledge transfer problematic, involving particularities of tacit knowledge applied to industrial context. In France, as many others European countries, important industrial progresses were accompanied by massive employment. In the 60's and 70's, we observed the emergence of technologies for high-speed trains, orbiting satellite or nuclear power plants. Many realizations by the world are testimonies of technical gestures performed by operators. Forty years later, industries as Electricité de France (EDF, French electrical energy producer) are now confronted to a massive retirement. By the way, this demographic transition is creating a sudden and urgent need for the transmission of tacit know-how. As made since decades, usual knowledge transfer is based on journeyman. This implicit intergenerational transfer needs one to one workers contacts and time to be achieved. Both conditions are now impossible to be satisfied, and this problem leads industries to solve differently the equation.

To consider a usual definition, the gesture is a form of non-verbal communication, implying necessarily a movement of a part of the body. There exist gestural codes in any work, in the manners of being, of behaving, of communicating, which are specific to each work environment. The topic of this paper is to approach the professional gesture as a tacit knowledge of industry but it could be approached as a part of an action, implying a communication between operator and machine.

Researches in ergonomics and work psychology on knowledge embodied in gestures impact different sectors of activities: tourism [START_REF] Sauvage | Formation au geste professionnel par multimédia[END_REF], automobiles and civil engineering [START_REF] Chassaing | Vers une compréhension de la construction des gestuelles avec l'expérience: le cas des "tôliers" d'une entreprise automobile[END_REF], rail [START_REF] Fernandez | Le corps, le collectif et le développement du métier: étude clinique d'un geste de métier à la SNCF[END_REF], aviation [START_REF] Bibliography Aubert | Transformer la formation par l'analyse du travail: le cas des peintres aéronautiques[END_REF], surgery [START_REF] Tomás | La transmission des gestes de métier en chirurgie cardiaque[END_REF], etc. This literature doesn't show any consensus about terminology for work gesture analysis: Chassaing (2006) uses the terms "gesture of work," or "gestural knowledge". Clot, and al. [START_REF] Clot | Le geste de métier : problèmes de la transmission[END_REF] speak of gesture at work; Aubert (2000) speaks only of "gesture". However, all agree in showing the difficulties to understand the knowledge underlying professional acts by their specificity. Indeed, this knowledge belongs to a class of very special skills, those that are "encapsulated in the action, hard to verbalize, closely related to the context" [START_REF] Leplat | Skills and tacit skills: a psychological perspective Applied psychology[END_REF][START_REF] Leplat | A propos des comprétences incorporées[END_REF]. We speak of knowledge embodied, that is to say, "worn and stored by the body" [START_REF] Bibliography Aubert | Transformer la formation par l'analyse du travail: le cas des peintres aéronautiques[END_REF], which involves all the senses. These are the "skillbased behavior" of Rasmussen [START_REF] Rasmussen | Skills, Rules and Knowledge; Signals, Signs, and Symbols, and Other Distinctions in Human Performance Models[END_REF].

In the automotive sector, with assembly tasks, and in the civil engineering sector, with tasks of formwork, K. [START_REF] Chassaing | Vers une compréhension de la construction des gestuelles avec l'expérience: le cas des "tôliers" d'une entreprise automobile[END_REF] attempts to show the organization of work gestures learned on the job, their development, structure and implementation. Partly reflected by the characteristics of professional skills, she characterizes a work gesture in four points. First, the gesture is a composition: it "requires a sensory, cognitive and motor activity". Second, the gesture is invested: it "is intentional and is oriented toward different goals: to the system, to itself, and to others". Third, the gesture is situated, "the work activity [...] is in a situation". Last, the gesture is built: it "is the product of a history, of a past, and continues to be expanded."

The purpose of this research is focused on the capture and transfer of professional knowledge, and not on the study of gesture as such. For this, we use theories of work analysis, but from a quite different perspective from Clot et al. Indeed, our observations and analysis focused not on the overall activity of an operator on his job, but on specific segments of his activity selected by the organization and based on: 1) their criticality from the perspective of the organization, and 2) the expertise involved from the operator. These segments are called "professional gestures" by the organization EDF. What is called in common language an operation (e.g. maintenance), a maneuver (e.g. a seal change) or broadly an activity, is called "gesture" by the training professionals because of the strong physical, manual component that covers the job of operators in nuclear power plants.

From our perspective, we define the "professional gesture" as the outward manifestation of an activity segment, carrying expert skills and guided by motives and goals. The notion of segment implies that the duration of this activity is limited in time. Following Russian Activity Theory [START_REF] Leontiev | Activity, Consciousness, and Personality[END_REF]Nosulenko & Rabardel, 2007;[START_REF] Rubinstein | Osnovy obshej psikhologii (Les fondements de la psychologie générale). Moscou: Edition "Utchpedgiz[END_REF] which is our main intellectual framework, studied gestures are intentional, following a motive, and are determined by a set of goals achieved through actions.

Our work investigates the use of digital video and activity elicitation to give a guideline and operational tools for the capture of knowledge embodied in professional gestures. The operators are equipped with video capture devices. They are also invited to describe the intentions (the goals) of their gesture. But the main problem for this situation is to understand the gesture itself and to know if the gesture can be captured and how. How can it be characterized? Which aspects of the gesture are determinant for a good outside perception? How can we access to the gesture's cognition? What is the tacit part in a professional gesture?

Material and method

This section presents a brief synthesis of the Multimedia Apprenticeship Platform (MAP) construction methodology (Le [START_REF] Bellu | Capter et transférer le savoir incorporé dans un geste professionnel[END_REF] and an overview of collected gestures. The MAP is a tool that allows novices to represent themselves a mental model of the gesture, hierarchically and sequentially structured. It illustrates the gesture in details, with the help of video sequences.

To capture

Agreement step

A semi-directive interview brings together the analyst in charge of the MAP creation and an operator. The hierarchy has identified the operator and appointed him for his skills and expertise. This first stage addresses three objectives: 1) inform the operator of the process to which he/she is committing; 2) arrive at a common mental model of the gesture by asking the operator to mention his/her goals, and 3) collect information that will enable to plan the capture (what, where, who, when). We present below a set of ten professional gestures selected by the EDF training branch for their rare (only a few times performed throughout a career and mastered by a handful of people) and/or critical (i.e. involving safety, security and/or performance of the company) features. This gestures corpus constitutes our work material for this study.

G1: Remote operation of a tap

The tap is controlled via a "button box" that the operator connects to the system after have made the control power cell operational.

G2: Manual setting of a tap

The operator activates a tap high located and controls the adjustments effects on a high dial located at this level.

G3: Diagnosis of an electrovalve malfunction

By manually activating the valve, the operator identifies adjustments to be made.

G4: Diagnosis of a valve failure

Faced with a valve failure case, the operator compares the current valve with the representation of a correct one.

G5: Control of a valve tightness

After a maintenance operation, the operator controls pressure and parallelism of a valve tightness made by other operators.

G6: Condenser closure

Seal installation and valve closing by calibration and checking of the position of a large condenser horizontally installed.

G7: Consignment of a pump

The operator isolates and drains pumps containing superheated water. A first operator identifies a range of key points and organs to be "consignmented" on a map. A second operator does operations and reports to the first through an informal dialogue phase.

G8: Rond of safety operator

Check of a functioning site.

G9: Tightening of a bolted ensemble

Seal changing and bolted ensemble restoration on a valve.

G10: Chargers switching

The operator switches chargers on a 380V electrical board. Two different operators have performed this gesture.

Video recording

The operator wears a subjective camera (subcam) [START_REF] Lahlou | Observing Cognitive Work in Offices[END_REF][START_REF] Lahlou | How can we capture the subject's perspective? An evidence-based approach for the social scientist[END_REF]) -focused on the operator's activity from his own point of view, and another external camera filming the context of the scene -a hand-held camera and/or a camera on a support. For the purpose of this research, a special subcam was designed to be embedded on the protective helmet of operators working in factory environments (Figure 1). 2. The next moment is the recording one. The operator must perform the real actions of its gesture in the most natural way possible. In the specific case of a collaborative gesture, each operator wears a subcam. The operator is asked to think aloud, using a goal-directed verbalization protocol (Le [START_REF] Bellu | Comment capter le savoir incorporé dans un geste métier du point de vue de l'opérateur ? Information Sciences for Decision Making[END_REF] to collect the operator's intentions. This is done while accompanying each of the goals with explanations answering the following questions: For what reason (why) and in what manner (how) is the gesture performed? "The operator is freely allowed to consider what is worthy of comment, in regard to the level of sophistication in the breakdown of his/her activity, as well as the moments in time where a specific action begins or ends." (Le [START_REF] Bellu | Comment capter le savoir incorporé dans un geste métier du point de vue de l'opérateur ? Information Sciences for Decision Making[END_REF]. Our experiments show that a goal-directed verbalization protocol aims to externalize tacit knowledge and access to the gesture's cognition. With the mental preparation just before, the operator has engaged his knowledge and has prepared his speech. By the way, the inevitable disruption of the gesture is reduced [START_REF] Ericsson | Thinking-aloud protocols as data: Effects of verbalization[END_REF][START_REF] Ericsson | Protocol Analysis: verbal reports as data[END_REF].

3. The just-after video-recording time is dedicated to an "on the spot" debriefing with the operator. Did he run into any problems? What were the specific components of this situation? What does stay the same, what doesn't usually change? By this questions, the analyst tries to access to the intraindividual diversity of the gesture.

To analyze and to formalize

Analysis: to study data One analysis of the gesture based on activity theory [START_REF] Leontiev | Activity, Consciousness, and Personality[END_REF][START_REF] Rubinstein | Osnovy obshej psikhologii (Les fondements de la psychologie générale). Moscou: Edition "Utchpedgiz[END_REF] and perceived quality theory [START_REF] Nosulenko | Psikhologia slukhovogo vospriyatia (La psychologie de la perception auditive). Moscou: Edition "Nauka[END_REF][START_REF] Nosulenko | Evaluation de la qualité perçue des produits et services : approche interdisciplinaire[END_REF], 2009) is performed in delayed time of the capture. Firstly, the recorded videos are evaluated considering their quality, quantity of information, correspondence to the capture protocol, etc. Secondly, the objective is to plan and to edit a version of a whole video of the gesture performance that puts together subjective and contextual points of view. This video-making must preserve a maximum of information about the gesture to be used as a guideline or the operator's commentary during the next step. For this analysis work, the analyst uses videos and others materials collected during previous steps. Thus, the analysis leads to develop initial hypotheses about the importance of the different operations and the intentional structure of the gesture according to activity's components: goals, sub-goals, tasks and operations [START_REF] Leontiev | Activity, Consciousness, and Personality[END_REF][START_REF] Rubinstein | Osnovy obshej psikhologii (Les fondements de la psychologie générale). Moscou: Edition "Utchpedgiz[END_REF].

Self-confrontation interview: accessing to the subject's subjective experience

The self-confrontation interview method [START_REF] Theureau | Le cours d'action, analyse sémio-logique: essai d'une anthropologie cognitive située[END_REF][START_REF] Cranach | Goal-directed action[END_REF] re-immerges the operator in his own activity by confronting him with the recorded gesture. The explanation interview [START_REF] Vermersch | L'entretien d'explicitation[END_REF][START_REF] Vermersch | L'entretien d'explicitation[END_REF], the crossed self-confrontation techniques [START_REF] Clot | Entretiens en auto-confrontation croisée : une méthode en clinique de l'activité[END_REF][START_REF] Mollo | Auto-and allo-confrontation as tools for reflective activities[END_REF]) and the re-situ interview [START_REF] Rix | Enregistrement en perspective subjective située et entretien en re situ subjectif: une méthodologie de constitution de l'expérience[END_REF]) are variations based on the same principle. The variant we are using shares with Cranach's method the fact of resting on the activity theory. Considering our previous practices of different situations [START_REF] Lahlou | Digitization and Transmission of Human Experience[END_REF][START_REF] Nosulenko | Man-technology interaction: some of the Russian approaches[END_REF][START_REF] Nosulenko | Psychological Methods for the Study of Augmented Environments[END_REF], we tend to be a little bit flexible in the formalization of our protocols. The main point is to collect reliable elements to be able to reconstruct the activity. The self-confrontation interview can be made not only with the mobilised operator, but also with another specialist of the considered gesture. This latter modus operandi has the advantage of more easily identifying the components of the knowledge: acquired expertise, lack of experience, practice. Verbal data recorded during that interview enable to emphasise components of the activity within the meaning of Russian activity theory.

Figure 2: G9 gesture MAP printscreen

Further analysis, final video-making and building of the MAP

Based on data collected during the self-confrontation interview, the analyst changes the first video to be the nearest of the operator's gesture mental model. He can add/delete video parts, add a voiceover, exploit the self-confrontation's soundtrack, add/delete text annotations on the video, adds/removes icons, etc. This further analysis gives rise to a Multimedia Apprenticeship Platform (MAP) is a mode of synthetic representation, which aims to present the most important components of the captured gesture according to the operator's point of view (Figure 2).

To validate and to use: final validation of the MAP resource

The MAP undergoes three levels of validation: -A business validation: to check if the MAP is consistent with the considered gesture. The operator begins this validation just after the capture, continues during self-confrontation and ends at this stage; -An institutional and hierarchical validation: to verify the compliance of the gesture in relation to the institutional repository; -An educational validation: to validate the relevance of educational content carried by the MAP.

After this validation process, trainers of the company's training department can use the MAP in their training sessions (Figure 3).

Characterize a professional gesture: a proposition of classification in eight points

All of the above professional gestures were included in a process of MAP building. For some of them the progression of the process was easy. It was more difficult for others. Considering the gesture itself, we distinguish eight characteristics, giving pieces of information for the "MAPpability" of gestures. This diagnosis leads to a classification of professional gestures in eight points. Each feature has four possible modalities. A bold font is used to indicate terms that were not encountered in our corpus of gestures.

A. Realism of the situation

This is the degree of spontaneity of the capture situation. At one pole are the most artificial situations: the action is performed but the result does not occur (e.g. training exercises); at the other pole, are the most natural situations (e.g. performance of a maintenance gesture in a working power plant).

Level 1: simulation of the only highlighted facts related to objects handled outside of the workplace. At this level, the context is only symbolized.

Level 2: the situation is re-created. The environment and work situation are simulated outside the natural workplace. Level 3: the real situation is prepared. The gesture is produced on the real workplace. Consequences are expected, and can therefore be anticipated. Level 4: this is the real life mode. The capture is made on-the-fly, not provided for.

B. Reproducibility-flexibility

The more there is a decision-making part in the gesture, the more the operator has latitude in the choice of the procedure adaptation. This raises the problem of capture: we capture a gesture that is truly "iconic"? Level 1: the gesture is made only once; it's enough to obtain the "example". Level 2: you have to do the same gesture several times to identify the example. Level 3: you must set several conditions to understand the mental model of the gesture. Level 4: however multiple instances of the gesture are performed, it can still happen something different. Each case is unique.

The degree of flexibility available to the operator can give some answers. For example, in the case of a monitoring (G8), there is a procedure to apply (e.g. every day, the scaffolding must be checked), but his flexibility is important. Operator organizes his course as he wishes. Another example is the G7 gesture in which a failure diagnosis is made following a leakage problem. At the opposite, the more the environment is controlled (e.g. G2: a procedure application), the closer we approach ordinary daylife gestures. Thus, more a flexibility exists, the less the gesture is exactly reproduced.

C. Length

A lot of questions exist of the relationship between perceived time and real time. Our observations show that the acceptable length of a film is about eight minutes to obtain a correct usable material. The length will be accentuated by the verbalization. Beyond, the perceived duration by the end-user apprentice is too long. This means that there would be levels of operability that pass through the cutting of a gesture in short sequences of less than ten minutes.

Level 1: cases of very short gestures (in the order of seconds). Level 2: cases of short gestures (in order of minutes). Level 3: gestures of about ten minutes. Despite the reproducibility of the gesture, we hesitate to repeat it, due to the time of capture. Level 4: gestures that we don't repeat the capture: several decades of minutes.

Perceived time is not only linked to real time, it includes perception of physical or intellectual effort. The longer the duration of the gesture is, the more it affects the ability to concentrate.

D. Granularity-accuracy

This is a notion of precision in gesture: rather thin ("fingertip") or rather wide (large arm movements). We notice that it is possible that the visual acuity of the observer plays a role in granularity definition.

Level 1: very thin. Level 2: thin. Level 3: wide. Level 4: very wide.

E. Movements-mobility in gesture

The mobility in the gesture requires sometimes body movements. It depends on the area, and if the involved system is or not distributed in space.

Level 1: no displacement. A single confined space. Level 2: multiple cameras/viewpoints are used to cover several fixed places. Example: in the G2 gesture, movement (in height) is sustained by the operator. He has no choice; it is a necessary action for the pursuit and realization of his gesture. Level 3: a mosaic of places, with a mobile camera to track the movements. The movement is a part of the gesture and it's something going on during this movement. Level 4: the gesture is a displacement. We must follow up. For example, in the gesture of monitoring (G8).

F. Collaboration and verbal interactions

The problem of communication and verbal interactions during the gesture is generally correlated with the inter-operator collaboration; this is what we call collaborative gesture.

Level 1: no verbal interaction. Level 2: from time to time interactions. The interaction must be anticipated; it is expected in the process. Example: a phone call to report the situation or to make inquiries. By the way, some verbal interactions may be present in a solitary gesture. Level 3: in situ and frequent interactions. The conversation is a part of the gesture and involves several operators in performing the gesture. The gesture is collaborative. Level 4: interactions strongly linked to the task. The gesture is conversation.

G. Visual salience

This characteristic indicates a visual component in the gesture realization.

Level 1: the gesture may be performed blind (e.g. as an answer at the phone). Level 2: some visual cues are necessary. Level 3: a visual control is required for the gesture performing. Level 4: a monitoring activity.

H. Haptic salience

This characteristic indicates an haptic component in the gesture realization.

Level 1: the gesture can take place hands in pockets. Level 2: the gesture only requires a few tactile landmarks. Level 3: touch is necessary for the gesture. Level 4: permanent touch.

Application scope of the capture method

Figure 4 provides an overview of the studied gestures profiles (G1-G10). It shows that most cases were covered by the experiments. On the 32 possible items (eight characteristics, each with four terms), six have not been encountered in the studied corpus. These are the terms notified in bold in the previous section. We have highlighted them by black circles in Figure 4. Gestures with profiles located as in the periphery of the diagram, such as G6 and G8, correspond to "complex gestures"; the MAP is difficult to obtain. Those located further inside, such as G1, G2 and G10 are the most MAPpable: these are "simple gestures". Between, it is recommended to dwell on the reading of characteristics. Figure 5 describes each of the profiles.

G1

G2 G3

G4 G5 G6

Figure 5 : Profiles of each captured gesture

Discussion, conclusion

Which limits for capturing and transmit professional gestures? This question asks whether all professional gestures can be captured, analyzed, and formalized to be transferred in the form of a MAP. Our experiments distinguish between simple and complex gestures.

In the presented work, each of the ten withheld gestures was studied. All of them provide a part of tacit knowledge. The eight proposed descriptors give a pertinent way to approach this tacit part. Intuitively, gestures with small values on their descriptors are easy to formalize, with a "controlled" tacit part.

Then the profile of each gesture, as shown in figure 5, is a rapid clue to synthetize the tacit component of a gesture.

We have also put forward that descriptors do not all have the same importance. Items as collaborative, decision-making and motor/visual characteristics are involved in determining the complexity of a gesture. They are present in the descriptors F (verbal interactions) and G (visual component of a gesture). High values for these two descriptors imply difficulties to capture, to analyze and to validate the MAP (the three steps of the methodology). High values on other descriptors lead to localized difficulties during the methodology application: during the capture for descriptors A (realism), D (granularity) and E (mobility), mainly for the quality of the recorded video; during the analyze for descriptor B (reproducibility); or during the video edition for descriptor C (length). Descriptor H (haptic component) is well apprehended due to the position of the camera on the head of the operator.

The visual component of a gesture (descriptor G, i.e. the fact that the operator needs his eyes to perform the gesture) is present with a high value in the majority of gestures: level 3 for G1, G3, G4, G6, G7 and G10; and level 4 for G8. But only G8 leads to real difficulties to formalize and to analyse; to an extent G6 also gives us some problems. This is due to the fact that G8 has a small value for haptic component (level 1) whereas other gestures have a higher level (3 or 4). In fact the ratio Descriptor G/Descriptor H seems to be more pertinent than simple values of descriptors. Nevertheless, considering this ratio is not enough. G4 has the same big value for ratio G/H, without the same problems to formalize. The area of all descriptors must be taken into account. While we have small values as 16 to 19 for G1, G2, G3, G4, G9 and G10, this area growths to 22 for G7, 24 for G8 and 27 for G6. Due to the interactions between operators (G6 and G7) or interactions with other people (G8), collaborative gestures lead also to difficulties for the formalization step. This collaboration must be G9 G7 G8 G10 synchronous (as for G6 with two operators working in the same time on the same material) or takes place during key times (as for G7 with two operators following individual specific goals).

Our work shows that a special attention must be considered on visual component (with G/H ratio); collaborative form and global descriptor area. These are three important clues to transfer tacit knowledge from operators to novices.

This repository of professional gestures could be used as a tool to classify other gestures, compare them to encountered cases, and thus ensure they can be transferred. Then, understanding their criticality leads to adapt the method and anticipate ahead difficulties. This work on the capture of many professional gestures makes the link with the tacit knowledge and its problems of capture and transmission. Considering operational acts, we obtain an approach of expertise. The limitations of our approach meet the limits of the distanced course with respect to the journeyman approach. Although this work remains in terms of observation, the complement given by the perceived quality theory enables to put forward intentions of the operator by verbalization techniques.

Figure 1 :

 1 Figure 1: The "hard hat" worksite version of the subcam.

Figure 3 :

 3 Figure 3: use of the MAP during training session

Figure 4 :

 4 Figure 4 : Overview of the captured gestures corpus


Jean-Baptiste Authesserre
email: jean-baptiste.authesserre@ims-bordeaux.fr

Rémi Mégret
email: remi.megret@ims-bordeaux.fr

Yannick Berthoumieu
email: yannick.berthoumieu@ims-bordeaux.fr

Automatic Estimation of Asymmetry for Gradient-Based Alignment of Noisy Images on Lie Group

Keywords: Asymmetric image alignment, noisy images, parametric motion estimation, gradient methods, Lie Groups

Many parametric image alignment approaches assume equality of the images to register up to motion compensation. In presence of noise this assumption does not hold. In particular, for gradient-based approaches, which rely on the optimization of an error functional with gradient descent methods, the performances depend on the amount of noise in each image. We propose in this paper to use the Asymmetric Composition on Lie Groups (ACL) formulation of the alignment problem to improve the robustness in presence of asymmetric levels of noise. The ACL formulation, generalizing state-ofthe-art gradient-based image alignment, introduces a parameter to weight the influence of the images during the optimization. Three new methods are presented to estimate this asymmetry parameter: one supervised (MVACL) and two fully automatic (AACL and GACL). Theoretical results and experimental validation show how the new algorithms improve robustness in presence of noise. Finally, we illustrate the interest of the new approaches for object tracking under low-light conditions.

Introduction

Parametric motion estimation is a fundamental task of many vision applications such as object tracking, image mosaicking, video compression and augmented reality. To recover the motion parameters, direct image alignment works by optimizing a pixel-based error measure between a moving image and a fixed-image called template. Gradient-based alignment uses more specifically a local optimization relying on the gradients of the error measure with respect to the motion parameters, usually applied in an iterative scheme. These gradients in turn depend on the gradients of the images themselves. Also known as template matching, this method was first introduced for optical-flow computation [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF]. Since then, many related approaches were proposed. [START_REF] Baker | Lucas-Kanade 20 years on: A unifying framework[END_REF] summarized four main classes of algorithms as Forwards Additive, Forwards Compositional, Inverse Additive, and Inverse Compositional. [START_REF] Mégret | The bi-directional framework for unifying parametric image alignment approaches[END_REF] extended this taxonomy of the methods in order to take into account recent approaches: the Efficient Second-order Minimization (ESM) algorithm [START_REF] Benhimane | Homography-based 2d visual tracking and servoing[END_REF] and the Symmetric Gradient Method (SGM) [START_REF] Keller | Fast motion estimation using bidirectional gradient methods[END_REF]. These methods symmetrically weight the gradients of both images when estimating the update of motion parameters, which yields faster convergence and improved robustness. Another contribution of [START_REF] Benhimane | Homography-based 2d visual tracking and servoing[END_REF] was to introduce Lie group parameterization in image alignment, which is a natural and efficient way to parameterize motion [START_REF] Vercauteren | Diffeomorphic demons: Efficient non-parametric image registration[END_REF], especially for rigid motion estimation.

The approches using symmetrically weighted gradients have been shown to be more robust [START_REF] Keller | Fast motion estimation using bidirectional gradient methods[END_REF][START_REF] Benhimane | Homography-based 2d visual tracking and servoing[END_REF] than the forwards and inverse approaches. Those methods rely implicitely on the assumption that the gradients of both images are equally reliable. In presence of images of different SN R (Signal to Noise Ratio), this assumption does not hold. To improve the robustess of alignment algorithms in that case, [START_REF] Authesserre | Asymmetric gradient-based image alignment[END_REF] introduced the Asymmetric Compositional method as an extension of the ESM approach. This approach is based on the optimization of a closed-form error measure that leads to weighting the gradients of both images asymmetrically according to a free parameter α that tunes the level of asymmetry. Tuning this α parameter produces a family of approaches which can yield better performance than the conventional methods.

However the automatic computation of α was not provided.

In this paper, we propose new approaches relying on an Asymmetric Compositional on Lie Group (ACL) formulation of image alignment. The new approaches provide an adaptive computation of α to handle asymmetric levels of noise. The paper is organized as follows :

• In Section 2, we formalize the image alignment problem as a parameter optimization procedure on a Lie Algebra. We introduce the ACL formulation as a generalization of existing algorithms.

• In section 3, we provide a theoretical analysis of the ACL objective function.

-Its relationship with the ESM [START_REF] Benhimane | Homography-based 2d visual tracking and servoing[END_REF]) optimization procedure and the effect of noise are highlighted.

-A first algorithm, the MVACL is introduced in the context of a Gauss-Newton optimization of the ACL objective function. This approach focus on the amount of noise only and computes α subject to the knowledge of the relative variance of noise in the two images (section 3.2).

• In section 4, we introduce new algorithms to compute α :

-The GACL and AACL approaches propose more general heuristics to compute α automatically (sections 4.1 and 4.2).

-To relieve the overhead for the computation of α, fast approximations of those methods are also proposed (section 4.3).

• In section 5 the state-of-the art and the new approaches performance in the context of image alignment under various noise conditions are evaluated.

• In section 6, we illustrate the interest of the new approaches for object tracking under low-light conditions.

Image alignment

Lie group parameterization of motion

Rigid motion models such as non degenerate affine motion for the Euclidian plane x ∈ R 2 and homography for the projective plane x ∈ P 2 (Bayro-Corrochano and Ortegón-Aguilar, 2007) can be parameterized using Lie Group. A Lie group P is a differentiable manifold structured by the composition operation (•). To any finite dimensional Lie Group is associated a Lie algebra P whose underlying finite dimensional vector space is the tangent space to P at the neutral element 0. Locally an increment δµ ∈ P around 0 can be bijectively reparameterized by an increment v ∈ P using the exponential map:

δµ(v) = exp(v), (1)
with the following properties:

exp(-v) = exp(v) -1 , (2) exp(αv) • exp(βv) = exp((α + β)v), (3)
for all α, β ∈ R.

With such a mapping we can solve for parameters in the algebra and get the corresponding element of interest δµ using eq. ( 1). This ensures to obtain δµ inside the group without having to enforce explicit constraints [START_REF] Vercauteren | Diffeomorphic demons: Efficient non-parametric image registration[END_REF], as opposed to using the more classic vector space embedding P ∈ R p , where p is the number of parameters to estimate in the vector δµ.

In the following, as in [START_REF] Bayro-Corrochano | Lie algebra approach for tracking and 3d motion estimation using monocular vision[END_REF][START_REF] Benhimane | Homography-based 2d visual tracking and servoing[END_REF][START_REF] Vercauteren | Diffeomorphic demons: Efficient non-parametric image registration[END_REF], we assume that the parameter space P forms a Lie group, which acts on image coordinates x through W. Moreover, we assume that the motion model satisfies group action properties [START_REF] Miller | Group actions, homeomorphisms, and matching: A general framework[END_REF]. This action has the following properties, which are related respectively to composition (•), inversion ( -1 ) and parameters of the identity transformation 0:

W(µ k • δµ, x) = W(µ k , W(δµ, x)), (4)
y = W(µ -1 , x) ⇔ x = W(µ, y), (5)
W(0, x) = x. (6)
Explicit expressions of the group action W and of the exponential map exp are provided in Appendix B for the case of homography, which will be used in the experiments.

Asymmetric image alignment on Lie Group

The goal of the algorithms presented in this paper is to align a template image T (x) to an input image I(x), where x = (x, y) T is a column vector containing the pixel coordinates. The alignment problem is solved iteratively.

We assume that we know at step k a coarse estimation of the warp parameters µ k . Aligning two images on Lie group is formalized [START_REF] Authesserre | Asymmetric gradient-based image alignment[END_REF] as estimating the vector v ∈ P such that the following discrepancy error is minimal:

E α (v) = i∈1..N e α,i (v) 2 (7)
where e α,i represents the error at each pixel x i belonging to a region of interest

R = (x 1 , . . . x N ): e α,i (v) =I(W(µ k • exp((1 -α)v), x i )) -T (W(exp(-αv), x i )). ( 8
)
The particularity of eq. ( 8) is to introduce a tuning parameter α ∈ [0, 1] corresponding to an asymmetric constraint imposed on the alignment process.

The α coefficient allows us to formulate existing approaches generically:

the particular cases α = 0 and α = 1 correspond to extending the classical approaches (Forwards Compositional [START_REF] Shum | Construction of panoramic image mosaics with global and local alignment[END_REF] and Inverse

Compositional [START_REF] Baker | Lucas-Kanade 20 years on: A unifying framework[END_REF])) to Lie Group parameterization.

We call these extensions FCL (Forward Compositional Lie) and ICL (Inverse Compositional Lie). The FCL approach has already been introduced under the name LIEMANIFOLD approach in [START_REF] Guangwei | Optimization on lie manifolds and projective tracking[END_REF]. The case α = 0.5 corresponds to the ESM (Efficient Second-order Minimization) algorithm presented by [START_REF] Benhimane | Homography-based 2d visual tracking and servoing[END_REF] under conditions discussed in [START_REF] Mégret | The bi-directional framework for unifying parametric image alignment approaches[END_REF][START_REF] Authesserre | Asymmetric gradient-based image alignment[END_REF]. Thus the asymmetric method consists in generalizing previous approaches to any α ∈ [0, 1]. Following this naming scheme we will denote in the sequel ACL (Asymmetric Compositional Lie) the asymmetric approach where α is arbitrarily set a priori.

Aligning two images consists in optimizing locally the error defined by eq. ( 7) and ( 8) at step k with respect to v. Once v is computed, the update rule provides a new estimation of the motion parameters:

µ k+1 = µ k • exp(v).
(9) [START_REF] Baker | Lucas-Kanade 20 years on: A unifying framework[END_REF] showed that the Gauss-Newton (GN) optimization has efficient convergence properties and a reasonable computational cost. Following most authors [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF][START_REF] Shum | Construction of panoramic image mosaics with global and local alignment[END_REF][START_REF] Keller | Fast motion estimation using bidirectional gradient methods[END_REF][START_REF] Vercauteren | Diffeomorphic demons: Efficient non-parametric image registration[END_REF], we will use this approach. In the sequel we denote by e α = (e α,i ) i=1..N the vector obtained by concatenating the pixel-wise errors e α,i . The Gauss-Newton optimization of the error function ( 7) yields:

vα = -(J α (0)) † e α (0) = -(J α (0) t J α (0)) -1 J α (0) t e α (0), ( 10
)
where t is the transpose operator and (J α (v 0 )) † is the pseudo-inverse of the jacobian matrix of the error vector e at 0. The jacobian matrix is defined as:

J α (w) = ∂e α (v) ∂v v=w (11)
This matrix can be expressed as the concatenation of the gradients J α,i (w)

of the pixelwise errors e α,i :

J α (w) = [J α,1 (w) t , . . . J α,N (w) t ] t with J α,i (w) = (1 -α)J I,i ((1 -α)w) + αJ T,i (-αw), (12)
where J I,i and J T,i are the jacobian matrices defined as:

J I,i (w) = ∂I(W(µ k • exp(v), x i )) ∂v v=w , (13)
J T,i (w) = ∂T (W(exp(v), x i )) ∂v v=w . ( 14
)

Theoretical analysis of the ACL objective function

We now introduce a theoretical study showing the usefulness of the generic ACL formulation to provide new insights into existing gradient-based approaches and the effect of noise. Considering successively the noiseless and the noisy cases, second-order expansion of the error function e α (v) is performed. This allows us to highlight in section 3.2 the trade-off that needs to be done between the minimization of both second-order structural terms and noise variance terms.

Let us consider two images corrupted by independant additive noises characterized by respective variances σ 2 I and σ 2 T :

I = I nf + ǫ I and T = T nf + ǫ T , (15)
where the indice nf indicates noise-free values. This yields the noisy jacobians:

J I (w) = J nf I (w) + J ǫ I (w) and J T (w) = J nf T (w) + J ǫ T (w), (16)
where J ǫ I (w) and J ǫ T (w) are the jacobian matrices obtained by concatenating the pixelwise gradients:

J ǫ I ,i (w) = ∂ǫ I (W(µ k • exp(v), x i )) ∂v v=w , and (17)
J ǫ T ,i (w) = ∂ǫ T (W(exp(v), x i )) ∂v v=w . (18)

Noise-free case

In this section we consider that the noise-free images are identical up to the true motion parameter vector v * :

∀x I nf (W(µ k • exp((1 -α)v * ), x) = T nf (W(exp(-αv * ), x)), (19)
which yields:

e nf α (v * ) = 0. ( 20
)
Under this assumption we can show that:

Theorem 1 (Second-order expansion under noise-free conditions at optimum v * ). For any α:

e nf α (v * ) = e nf α (0) + 1 2 (J nf I (0) + J nf T (0))v * + O(||v * || 3 ). ( 21
)
Proof. We use here the lemmas presented in Appendix A. According to (A.1), we have to second-order:

e nf α (v) = e nf α (0) + 1 2 (J nf α (0) + J nf α (v))v + O(||v|| 3 ). ( 22
)
Moreover, for the true parameter v * , and using (A.13), we have:

(J nf α (0) + J nf α (v * ))v * = (J nf I (0) + J nf T (0))v * + O(||v * || 3 ) (23)
Plugging ( 23) into ( 22) yields (21).

Note that e α (0) is independant of α. [START_REF] Benhimane | Homography-based 2d visual tracking and servoing[END_REF] initially proposed the ESM procedure consisting in plugging the jacobian J ESM into the equation ( 10) as a second-order minimization of the forward objective function (α = 0 in (8)) where:

J ESM = 1 2 (J I (0) + J T (0)). ( 24
)
Their proof was based on the assumption (19). Under the same assumption, equation ( 21) generalizes this result by revealing that the ESM optimization yields in fact a second-order minimization of any ACL objective function, with FCL and ICL objective functions as special cases.

In presence of noise

Under noisy conditions eq. ( 23) cannot be applied on the noisy jacobian J α and holds only for the noise-free part J nf α . Furthermore, during the optimization procedure we have only access to the noisy image jacobian J I (0) and J T (0). Thus, we are looking for the α value that produces a jacobian J α (0) as close as possible to the jacobian J nf ESM .

Theorem 2 (Discrepancy between J α (0) and the ideal jacobian

J nf ESM ). J nf ESM = J α (0) -(A α + B α ), (25)
with:

A α = ( 1 2 -α)J nf ⊖ (0), B α = (1 -α)J ǫ I (0) + αJ ǫ T (0), (26)
where :

J nf ⊖ (0) = J nf I (0) -J nf T (0). ( 27
)
Proof.

J nf ESM = J nf α (0) -( 1 2 -α)J nf ⊖ (0) = J α (0) -( 1 2 -α)J nf ⊖ (0) -((1 -α)J ǫ I (0) + αJ ǫ T (0)) (28)
Equation ( 25) highlights the presence of two kinds of corrupting terms A α v * and B α v * when plugged into eq. ( 21).

Term of structure. The term A α v * corresponds to second-order terms of the noise-free error e nf α (v * ) at optimum as it can be shown by plugging (A.13) into (A.4):

J nf ⊖ (0)v * = (J nf I (0) -J nf I (v * ))v * (A.4) = O(||v * || 2 ). ( 29
)
Using the ESM algorithm allows us to compensate them and therefore reach a second-order approximation in the noise-free case.

Term of noise. The term B α v * corresponds to first-order contribution of noise. Assuming the noises on I and T are independant of respective variance σ 2 I and σ 2 T and of zero mean, the total variance of B α v * is proportional to:

var(B α v * ) ∝ (1 -α) 2 σ 2 I + α 2 σ 2 T (30)
Minimizing ( 30) with respect to α defines a new Minimal Variance ACL approach denoted MVACL:

α MVACL = σ 2 I σ 2 I + σ 2 T . (31)
This α value has the following properties: it yields an ESM behaviour for symmetric amount of noise (σ I = σ T ) and unidirectional FCL behaviour (resp. ICL behaviour) when σ I ≪ σ T (resp. σ I ≫ σ T ).

Several typical cases can be considered:

• The SNR is high and the initialization is close to the optimum (||v * || 2 ≪ ||v * ||): the terms A α and B α are small and do not influence the alignment process, the FCL, ICL, ESM and MVACL perform equivalently.

• The SNR (signal to noise ratio) is high and the initialization is far from the optimum (||v * || is large): the term A α dominates B α and the optimal α is near 0.5, as used by the ESM.

• The SNR is low, but the noise is symmetrically distributed (σ I ≈ σ T ) on I and T , then the variance of B α v * is also minimum for α = 0.5, which is used by both the ESM and the MVACL algorithms.

• The SNR is low and asymmetrically distributed σ I ≫ σ T (resp. σ I ≪ σ T ) and the initialization is close to the optimum :

B α dominates A α ,
and the optimal α is the one that minimizes the variance of B α as estimated by the MVACL. For very asymmetric noise levels, one of the images is far more corrupted than the other. The optimal α value can reach the bounds 0 if T is the image strongly corrupted (resp. 1 if I is the image strongly corrupted), which is used by the FCL (resp. ICL) approach.

• The SNR is low and the initialization is far from the optimum : both terms A α and B α are not negligible and should be taken care of. In that case it is difficult to predict which algorithm is more efficient between the MVACL and the ESM.

It follows from the previous analysis that the ESM and the MVACL provide the same result for symmetrical levels of noise, but that in other cases, each one assumes one of terms A α or B α is negligible. Those remarks are summarized in table 1 and will be discussed further in the experimental section 5.2. The MVACL furthermore requires the knowledge of the noise variances. We propose in the next section to take a different perspective on this problem by using a geometrically motivated solution for the fullyautomatic estimation of α.

Fully-automatic estimation of α

Authesserre et al. (2009) have shown that the asymmetric approach can outperform other approaches in terms of robustness to noise by choosing an appropriate α. In the robotic community [START_REF] Marey | Analysis of classical and new visual servoing control laws[END_REF] propose a similar approach for designing command law in visual servoing. In both cases no rule was presented to compute the α-value. In this section we propose two methods based on an automatic computation of this coefficient.

4.1. Method 1 : Geometric ACL (GACL)

In the following, we denote by F α the affine subspace passing through e α (0) and parallel to the span of the columns of J α (0). The elements of F α can be obtained by the linearized error:

ẽα (v) = e α (0) + J α (0)v. ( 32
)
We also denote by vα the Lie algebra increment estimated by plugging J α (0) into eq. ( 10).

The geometric interpretation of the Gauss-Newton algorithm is that the increment vα is chosen such that êα = ẽα (v α ) corresponds to the orthogonal projection of the origin O of the error space onto the affine subspace F α .

Thus, the different ACL methods can be distinguished by the choice of the subspace F α onto which O is projected. This is illustrated in fig. 1 for the case of a motion model with one degree of freedom. The α value of the ACL approach can be seen as a confidence value on the directions (in the error space), of the subspaces F 0 and F 1 . It should favor the directions that lead closer to the origin O of the error space. We propose to compute α by using the distances of O to both subspaces F 0 and F 1 .

Considering the plane passing through the points ê0 , ê1 and O (see fig. 2), we denote by P the orthogonal projection of O onto the line passing through ê1 and ê0 , by l the distance between ê1 and ê0 and by d 1 (respectively d 0 )

the distance between O and ê1 (respectively ê0 ). Using the Pythagorean theorem in (O, P, ê0 ) and in (O, P, ê1 ), we obtain:

α GACL = d 2 0 -d 2 1 2l 2 + 1 2 = < ê0 |(ê 0 -ê1 ) > ||(ê 0 -ê1 )|| 2 , ( 33
)
where < .|. > (respectively ||.||) stands for the regular scalar product (respectively the Euclidean vector norm) in R N . The α GACL value favors the subspace nearest to O. If both subspaces are equally near to O, the α GACL value is equal to 0.5, and the behaviour of the ESM approach occurs. This method in its simplest form requires computing both ê0 and ê1 to estimate α GACL which is quite costly.

Method 2 : Analytic ACL (AACL)

The AACL algorithm considers the linearization of the pixelwise residual eq. ( 8) around v = 0 (cf. eq. ( 32)). An approximation of the complete residual error is obtained by plugging eq. ( 32) into eq. ( 7):

E α (v) ≈ ||e α (0) + J I (0)v|| 2 -2α(e α (0) + J I (0)v) T J ⊖ (0)v + α 2 ||J ⊖ (0)v|| 2 , ( 34
)
where J ⊖ = J I (0) -J T (0).

Minimizing this residual with respect to α for a fixed v yields the confidence value used by the AACL algorithm:

α AACL (v) = < ẽ0 (v)|(ẽ 0 (v) -ẽ1 (v) > ||ẽ 0 (v) -ẽ1 (v)|| 2 , ( 35
)
where ẽα (v) is defined by equation ( 32).

Notice the similarity between the expressions (33) and ( 35). In the first case, α is obtained by projecting O orthogonally onto the affine line (ê 0 , ê1 ).

In the second one, O is projected orthogonally onto the line (ẽ 0 (v), ẽ1 (v))

instead. For the AACL method, a fixed v has to be set before using eq. ( 35). In the sequel, we note AACL M the analytic AACL approach using the method M for computing v, where M can be the FCL approach (α = 0), the ICL approach (α = 1) or the ESM approach (α = 0.5).

Summary of the algorithms and fast approximations

The state-of-the-art gradient-based approaches and the new ones can be seen as particular cases of the ACL formulation using different values of α for the computation of the jacobian matrix eq. ( 12). The different methods considered in this paper are summarized in the table 2. The main steps of the algorithm scheme is summarized in table 3.

ACL Methods Non-adaptive to noise

Adaptive to noise Fixed-α Supervised-α Automatic-α 31) eq. ( 33) eq. ( 35)

FCL ICL ESM MVACL GACL AACL M α 0 1 0.5 α MVACL α GACL α AACL (v) eq. (
Table 2: The different asymmetric approaches. For the AACL approach, M stands for the method (FCL, ICL or ESM approach) used to set v in equation ( 35).

For step 2b), if the automatically computed α is such that α > 1 (resp. is such that α < 0), we arbitrarily set it to 1 (resp. 0) yielding the behaviour of the ICL (resp. FCL) approach. All approaches are based on the computation of the Jacobian matrices J I and J T . Implementation issues are tackled in appendix Appendix C.

As noted by [START_REF] Baker | Lucas-Kanade 20 years on: A unifying framework[END_REF], steps 2a, 2b and 2c can be skipped for the ICL approach (α = 1). Moreover, for this method, the Iterative scheme of the ACL approaches 1. Precomputation: (a) Compute the template jacobian J T (0) (eq. ( 14)), (b) Estimate α for the MVACL, F -GACL and F -AACL (eq. ( 31), ( 33) and ( 35)). 2. Iterate until convergence:

(a) Compute the current error e α (0) (eq. ( 8)) and the image jacobian J I (0) (eq. ( 13)), (b) Only for AACL and GACL approaches, compute α (eq. ( 35) or ( 33)), (c) Compute the asymmetric jacobian J α (0) (eq. ( 12)), (d) Solve for v (eq. ( 10)), (e) Update parameters µ k+1 (eq. ( 9)). pseudo-inverse of the jacobian J α=1 (0) = J T (0) can be precomputed. The ICL algorithm is thus the most efficient of the ACL approaches in terms of computational time per iteration.

The AACL and GACL algorithms may introduce a significant overhead in terms of computational time for the computation of α (step 2b) if they are run on a sequential processor. To improve the efficiency, we propose estimating α only once during the first iteration. This yields an approximation of the AACL and GACL denoted F -GACL (Fast-GACL) and F -AACL (Fast-AACL). The fast new approaches only introduce an additional computational effort for computing α during the first iteration compared with fixed α-ACL approaches (except for the ICL approach). This computational cost is called E-CC (for Extra Computational Cost) in the sequel. Moreover, we denote by P-CC (Pre-Computational Cost) the computational effort induced each time the image used as the template T changes (step 1) in table 3) and by I-CC (iterative computational cost) the standard cost of each iteration of a fixed-α-ACL algorithm (step 2a), 2c) and 2d) in table 3). The table 4 summarizes the computational time of the different approaches obtained with a Matlab implementation on a Intel(R) Core(TM)2 duo CPU 3.0 GHz with 4 GB of RAM. The number of pixels N of the template is 10000. The number of parameters for the homography is p = 8. The E-CC for the fast approaches takes approximately a fifth of the computational time of one iteration of the ESM. Thus the additional computational cost for computing α with the fast approximations of the GACL and AACL is very small when several iterations are required for convergence. The next section will show in which situations this cost is counterbalanced by an increased robustness of the estimation.

Experimental validation

The performance assessment uses a methodology inspired by the benchmark proposed by [START_REF] Baker | Lucas-Kanade 20 years on: A unifying framework[END_REF]. V ar(ǫ) . The RMS point error is defined as the root mean square error between the test points and the point locations obtained from the motion parameter estimated by an alignment algorithm in the reference image coordinates.

We will consider two main performance criteria defined by [START_REF] Baker | Lucas-Kanade 20 years on: A unifying framework[END_REF]:

-Average frequency of convergence: percentage of tests where an algorithm converged to the correct estimate (RMS Point Error < 1 pixel), -Average rates of convergence: for tests that converge for all methods, the average RMS point error is plotted against the algorithm iteration number.

In the sequel, the results are obtained by averaging the performances

Influence of the noise asymmetry

To simulate controled noise asymmetry, noise is parameterized by a total variance σ 2 , and an asymmetry coefficient β ∈ [0,1]. The total variance σ 2 is computed from a given global SN R in order to average results with images (figure 3) of the same quality. The amount of noise is allocated to I and T as follows:

σ 2 I = (1 -β)σ 2 , σ 2 T = βσ 2 , with β ∈ [0, 1]. ( 36
)
where σ I (resp. σ T ) is used as noise standard deviation to corrupt the image I (resp. the template T ) with an additive centered white Gaussian noise.

Six configurations are considered to evaluate the comparative performances for various typical situations:

• Exp.0 : near initialization (Point Sigma σ p equal to 6 pixels) and high SNR (Total SNR=15 dB)

• Exp.1 : near initialization (Point Sigma σ p equal to 6 pixels) and medium SNR (Total SNR=10 dB)

• Exp.2 : near initialization (Point Sigma σ p equal to 6 pixels) and low SNR (Total SNR=5 dB)

• Exp.3 : far initialization (Point Sigma σ p equal to 12 pixels) and high SNR (Total SNR=15 dB)

• Exp.4 : far initialization (Point Sigma σ p equal to 12 pixels) and medium SNR (Total SNR=10 dB)

• Exp.5 : far initialization (Point Sigma σ p equal to 12 pixels) and low

SNR (Total SNR=5 dB)

The results of the experiments 0-5 are shown on figure 5 and discussed in the sequel.

Conventional algorithms

The first column of the figure 5 shows the results of the FCL, ICL, ESM and MVACL algorithms. As mentioned in the theoretical analysis (section 3.1), the ESM provides better results when β is near 0.5 (see exp. 0-5), or when the total SN R is high (Exp. 0 and 3). By using only the gradients of I (resp. T ), the FCL (resp. ICL) yields only the best performance for β near 1 (resp. 0) or when the image T (resp. I) is sufficiently corrupted (Exp. 2, low SN R).

MVACL approach

As it can be seen on the first column of the figure 5, the MVACL provides performance similar to ICL for β = 0 that evolves smoothly to be similar to ESM for β = 0.5 and to FCL for β = 1. On the one hand, the MVACL yields a better behaviour than the standard approaches under low SN R conditions (Exp.2 and 5) because in that case the MVACL appropriately averages the term of noise B α of the equation ( 25). On the other hand, this approach is outperformed by the ESM under high SN R conditions (Exp. 0 and 3).

Indeed, by only taking into account the noise variance for computing α, the MVACL optimally weights noise-jacobian terms at the expense of taking into consideration the term A α , which is in that case not negligible compared to the term of noise B α .

Conventional approaches vs MVACL

Automatic approaches Fast Approximations

Exp.0

Influence of the choice of v for the AACL approaches

In the following we evaluate the new approaches. The second column of figure 5 shows the different behaviors of the AACL FCL , AACL ICL , AACL ESM and the GACL algorithms in terms of frequency of convergence with respect to image noise asymmetry. The ESM performances are also plotted as reference. We observe that:

• For any Point Sigma and Total SNR and for any β, the AACL M approach using the method M to initialize v in ( 35) always provides at least as good results as the method M itself (Exp.0-5).

• The performance of the AACL M is correlated to the performance of the method M itself. Thus, for situations where the FCL (resp. ICL) works well, such as in Exp.2 and and in Exp.5 for β near 1 (resp. near 0), the AACL FCL provides better results than the conventional methods and other AACL approaches by improving the performances of the FCL (resp. ICL) approach. In all other cases, the AACL ESM provides better results.

AACL vs GACL

For any β value and for any global SN R, the GACL and AACL ESM yield similar results. Moreover, they always provide the best results (Exp. 0, 1, 3 and 4) or near the best results (Exp. 2 and 5). The AACL FCL and AACL ICL are only more robust in very asymmetric cases (Exp.2 and 5) where the noisy image is not the one used for computing the gradients allowing the estimation of v. For all these reasons, we consider in the sequel only the GACL and AACL ESM approaches.

Performance of the fast approximations

The figure (5), third column shows the different behaviors of the ESM, F -AACL ESM and F -GACL algorithms in terms of frequency of convergence with respect to image noise asymmetry. It can be seen that the fast approximations produce the same behaviors as the exact AACL and GACL approaches. However, the fast approximations provide slightly worse results than the exact approaches, particularly for quasi symmetric amount of noise (β between 0.35 and 0.65) where they are outperformed by the ESM approach (see Exp. 2, 4 and 5). The fast approaches are thus a good choice for aligning a noisy image to another noisy image, but of greater quality (which in practice takes place for example in image mosaicking [START_REF] Pham | Performance of optimal registration estimators[END_REF] where a noisy image is aligned to the current mosaic).

To illustrate this, fig. 6 shows the average rate of convergence (Point Sigma σ p equal to 6 pixels) and average frequency of convergence with respect to σ p for a noisy image (SN R I = 10 dB) and a high quality template (SN R T = 100 dB). For legibility, only the fast approximations and the stateof-the art approaches are plotted. The fast new approaches outperform the state-of-the-art approaches in terms of frequency of convergence and convergence rate. The improvement in convergence rate counter-balances the E-CC introduced by the computation of α during the first iteration. Thus, the new approaches are at least as efficient as the best reference algorithms while better handling noisy images. Thus, for an equivalent computational complexity and according to the repartition of noise on images we can provide the following recommandation:

• In quasi noiseless condition or in symmetric noise condition (SN R I ≈ SN R T ), the ESM algorithm would provide the best results among the studied approaches.

• In very asymmetric noise condition SN R I ≫ SN R T (resp. SN R I ≪ SN R T ), the FCL algorithm (resp. ICL) would provide the best results among the studied approaches.

• In the other cases, the AACL ESM and GACL would be prefered if the best robustness is required. Trading off for computational performance, the F -AACL ESM or F -GACL would be chosen in presence of strong corruption with asymmetric noise levels.

Object tracking under low-light conditions

In low-light conditions, an optical imaging system produces bad quality images (cf. figure 7) that can be modeled according to [START_REF] Alter | An intensity similarity measure in low-light conditions[END_REF] : the observed number of photons at one pixel is drawn from a Poisson distribution whose parameter is proportional to the average received intensity. Thus, tracking an object using gradient-based approach becomes a challenging task because the Poisson noise can severely corrupt the gradients of the images.

In order to improve the tracking performance for this kind of data, one would try to lower the noise on the template by averaging several registered frames. This approach yields an asymmetric image alignment problem where the current image I is registered to the template T of higher quality. We now evaluate the usefulness of the proposed algorithms in such a context.

Corpus and ground truth design

We considered two sequences of 40 and 32 images captured with the same imaging device (samples are presenred on figure 7). In the first sequence one object of interest is selected and three are selected in the second sequence. This yields 136 independent observations of real shot noise associated to lowlight imaging. For each sequence and object, a ground-truth is generated by using the following protocol: • the object of interest is selected in the frame 1 of the sequence (which plays the role of the template image T ) through the use of a rectangular bounding-box. This location defines the true parameter μ(1) of the position of the object in the first image.

• The ESM algorithm is run to align image t+1 and image 1 by initializing alignment parameter with μ(t) and by using 200 iterations which allows the algorithm to estimate parameter μ(t + 1). The ESM algorithm is used here because it has been shown to be the more efficient in presence of symmetric noise. The motion model used is 2D translation. We checked manually frame by frame that the ESM produced reasonable motion parameters.

The generated ground-truth μ is approximative. We will not use it to do accuracy evaluation. However, we can use it to evaluate convergence rate and frequency of convergence of the different algorithms with a sufficiently large error threshold.

Evaluation methodology

The following results are obtained by averaging performance obtained for many different motion initializations for each image and for each object of interest. In order to do this, we use a benchmark similar to the one presented in section 5. First, the corners associated to the object of interest in the current image t are computed by using the parameters μ(t). A perturbation δx drawn randomly from a zero-mean normal distribution of standard deviation σ p = 10 is added to the spatial location x of each corner. The new locations of the corners yield the initial parameters µ 0 I (t) used to initialize the different alignment algorithms.

Two setups are studied here :

• Exp.6 : the algorithms are run using only the first image as a template.

The level of noise is thus symmetric.

• Exp.7 : each algorithm generates its template by averaging the five first frames of the video, after compensating the images in the same coordinate frame by using the estimated motion parameters.

For each frame, 20 different motion initializations are generated which yield more than 2000 tests for each setup.

Results

Figures 8 show the The ESM algorithm and the new approaches ACL provide better results than the more conventional approaches FCL and ICL for the Exp.6 which confirms on real data the good behavior of such approaches under symmetrical level of noise.

The use of an averaged template (Exp.7) improves the performance of all approaches that use the gradients of T . This improvement yields the approaches AACL, GACL and their fast approximations to outperform the other approaches in terms of convergence rate and frequency of convergence.

Remark on the accuracy of the approaches. In figure 8, after convergence, using only the first frame as the template (Exp. 6) has lower RMS point error than generating the template with the first five frames (Exp.7). The difference in accuracy observed under symmetric or asymmetric conditions are around 0.5 pixel. Because the images are very corrupted, it is difficult to check manually that the ground truth provides the position of the object with such a precision. Thus, there are two possible explanations for the difference in RMS Point Error at convergence :

1. By reducing the noise injected to the system in the Exp.7, averaging yields results which are more accurate than without the averaging (inaccurate ground-truth).

2. Because of a non perfect motion estimation, frame averaging in Exp.7 introduces a slight blur on the template which yields a decrease of the accuracy of the approaches (blurred template).

This analysis reveals that, when averaging frames, a trade off has to be made between noise and blur reduction.

The averaging process was only introduced to illustrate the potential of the ACL formulation to register images in presence of noise level asymmetry.

The study of using many frames to generate a high quality template is an interesting perspective, which is out of the scope of this paper.

In both Exp.6 and Exp.7 the automatic-ACL approaches provide as accurate results as the best of the state-of-the art approach, while improving robustness and convergence rate under asymmetric conditions, thus showing their ability to estimate a relevant α on such real datasets.

Illustration of tracking

To illustrate the principle of tracking using an average template, figure 9 presents the results of the approaches FCL, ICL, ESM and F -AACL ESM (the different heuristics AACL, GACL, F -AACL ESM and F -GACL ESM approaches provide similar results on those data) under real tracking conditions. As in Exp.7, each algorithm generates its template by averaging the five first frames of the video, after compensating the images in the same coordinate frame by using the estimated motion parameters. The motion parameters are initialized in each frame by those estimated in the previous frame. Each algorithm performs 30 iterations per image. The FCL approach fails to track the object of interest. The approach ESM looses the object of interest after the frame 20. The ICL approach accumulates tracking error and fails to track after the frame 25. The F -AACL ESM approach is able to track the object during the whole sequence.

Such an illustration shows under a realist context, the potentiality of the new Asymmetric approaches.

Conclusion

In this paper we used the Asymmetric Composition on Lie Group (ACL) formulation proposed by [START_REF] Authesserre | Asymmetric gradient-based image alignment[END_REF] to provide new gradientbased image alignment methods that compute automatically the asymmetry parameter α. A theoretical analysis allowed us to identify that in presence of noise, the role of the asymmetry parameter α is to perform a trade-off involving two terms : the deterministic structural term and the random noise term. Three new approaches have been proposed, i.e. the MVACL, GACL and AACL. The MVACL approach is more of a theoretical interest as it ignores the structural term completely, in favor of the minimization of noise variance. The two other approaches GACL and AACL are heuristics based on a geometrical and an analytical interpretation of the problem, which were shown to estimate relevant values of α, yielding improvement in the robustness to asymmetric noise. Those heuristics have practical interest, since they are fully automatic and reach comparable performance as the MVACL for asymmetric noise, even though the later requires the knowledge of noise variances. An interesting perspective for a future work, would be to take into account this information to further improve the performances.

The new methods introduce a computational overhead because of the need to compute α. To alleviate the computational burden, fast approximations of the new methods have been introduced, which have comparable computational complexity as fixed-α state-of-the-art methods, but with better robustness in case of asymmetric noise levels. with J α,i (0) the jacobian matrix and H α,i (0) the Hessian matrix of e α,i at 0:

H α,i (w) = ∂ 2 e α,i (v) ∂v 2 w (A.3)
We also have:

J α,i (v) = J α,i (0) + v t H α,i (0) + O(||v|| 2 ) (A.4)
Thus, by plugging (A.4) into (A.2), we obtain (A.1). This linearization is valid in noise-free as well as in noisy cases.

Lemma (Jacobian equivalence along v * ).

J nf I ((1 -α)v * )v * = J nf T (-αv * )v * (A.5)
Proof. Using directional derivative along v * , the assumption (20) and the exponential map properties (2) and (3), we have: with the special cases:

J nf I,i ((1 -α)v * )v * = ∂I nf (W(µ k • exp((1 -α)v * + tv * ), x i )) ∂t
J nf I (v * )v * = J nf T (0)v * (A.8) J nf T (-v * )v * = J nf I (0)v * (A.9)
where Tr is the trace of a matrix. The G m are the vector basis of the tangent space to the Lie Group manifold around the identity parameter µ = 0.

The G m are chosen such that they convey a geometrical meaning. The generators associated to translations are : Those gradients can be implemented using the Söbel operator.

2. M W is a matrix corresponding to the Jacobian of the warp:

M W = ∂W(µ, x i )) ∂µ 0 (C.5)
3. M L corresponds to the Jacobian of the Lie algebra reparameterization:

M L = ∂ exp(v) ∂v 0 (C.6)
M W and M L can be precomputed before running the alignment algorithm.

M T can also be precomputed. That is why the inverse approach using only gradient of the template is computationally the most efficient of the ACL approaches. M I (and consequently J I,i (0)) has to be computed at each iteration.

Particular case of homography. For the homography model

M W =   x T i 0 -x i x T i 0 x T i -y i x T i 0 0 0   ∈ R 3×9 (C.7) M L = [[G 1 ] v ... [G 8 ] v ] ∈ R 9×8 (C.8)
where [G m ] v is the matrix G m reshaped as a vector (in row major order) and

x i are homogeneous image coordinates x i = [x i y i 1] T ∈ P 2 .

Figure 1 :

 1 Figure 1: Geometric interpretation of the different approaches using a Gauss-Newton optimization for one parameter motion: O is projected orthogonally onto a particular affine subspace F α (in this case, affine line for a model of p = 1 parameter). The projection of O onto F α is noted êα .

Figure 2 :

 2 Figure 2: Generic interpretation of the α GACL value: The distances between the point O and the subspaces F 0 and F 1 are estimated. Then α GACL is chosen to favor the nearest subspace to O.

  Random homographies are generated by adding spatial Gaussian noise of standard deviation σ p (called Point Sigma) to four canonical point locations in a reference image I ref . The new locations are called test points. Using these homographies, the image I ref is warped onto the template image T ref . I ref (resp. T ref ) is then corrupted with additive gaussian noise: I = I ref + ǫ I (resp. T = T ref + ǫ T ). Noise level is characterized by the SNR (Signal to Noise ratio) between the noise-free image and the image of noise: SN R dB = 10 log 10 E(I 2 )

Figure 3 :

 3 Figure 3: Images used for the experiments. First raw: The noise-free image (left) is used for extracting the template image. Image I is then obtained by adding gaussian noise to the noise free image. The noisy images (right) are shown with respective SNR : 15 dB, 10 dB and 5 dB. Second raw: other images used for the experimentations. Images A and E come from the INRIA Learning and Recognition in Vision (LEAR) dataset (http://lear.inrialpes.fr/data). Images B and C come from the benchmark of[START_REF] Baker | Lucas-Kanade 20 years on: A unifying framework[END_REF]. Image D was synthetized using an openGL implementation.

Figure 4 :

 4 Figure4shows the behaviours of the ACL approaches, using a fixed value for α, in terms of frequency of convergence with respect to image SN R, when aligning an almost noise free template to a noisy image. As mentioned by[START_REF] Authesserre | Asymmetric gradient-based image alignment[END_REF], the best performance is obtained by a compromise between the use of the gradients of both T and I, and the weighting of the influence of noise. Thus, when the images are noise free (high SNR), the FCL (α = 0) and the ICL (α = 1) provide identical results and the ESM (α = 0.5) outperforms these two methods. However, when the noise level increases on I, the performance of the ESM decreases. When the SN R of I

Figure 5 :

 5 Figure5: Average frequency of convergence with respect to noise asymmetry for the different approaches. Exp. 0: σ p = 6 pixels, SNR=15 dB. Exp. 1: σ p = 6 pixels, SNR=10 dB. Exp. 2: σ p = 6 pixels, SNR=5 dB. Exp. 3: σ p = 12 pixels, SNR=15 dB. Exp. 4: σ p = 12 pixels, SNR=10 dB. Exp. 5: σ p = 12 pixels, SNR=5 dB. Corresponding numerical values for characteristic β values are presented in table 5.

Figure 6 :

 6 Figure 6: Average Frequency of convergence and average Convergence Rate for the ESM, ICL, FCL, F -AACL ESM , F -GACL algorithms in presence of image noise. A SNR of 10 dB has been used on the image and 100 dB on the template.

Figure 7 :

 7 Figure 7: Samples of low-light image.

Figure 8 :

 8 Figures 8 show the performance of the different algorithms in terms of mean RMS Point Error and frequency of convergence as a function of the iteration number for the Exp.6 and 7. The mean RMS Point Error corresponds to the mean euclidean distance between the four corners of the region of interest predicted by the tested algorithm and those of the ground truth.For the frequency of convergence, the convergence threshold is 4 pixels in terms of RMS Point Error.

Figure 9 :

 9 Figure 9: Tracking performance of the FCL, ICL, ESM and F -AACL ESM approaches on a video captured under low-light conditions. The first five images are used to build an average template. The dashed line rectangle corresponds to the ground truth location of the object. The solid line rectangle corresponds to the tracking results of the different algorithms after 30 iterations.

t=0 =

 t=0 ∂T nf (W(exp(-αv * + tv * ), x i )) ∂t t=0 = J nf T,i (-αv * ) ∂(-αv * + tv * ) * )v * = J nf I ((1α)v * )v * = J nf T (-αv * )v * (A.7)

  applying (B.4) to G 1 (resp. G 2 ) we obtain the standard 3 × 3 matrix of a translation along the x-axis (resp. y-axis). The generators associated respectively to isotropic dilation about the origin and rotation about the origin are: to the projective transformations induced by the parameters of the line at infinity are : Jacobian computationIn this section we tackle implementation issues about the Jacobian matrix associated to an image.The Jacobian Matrix associated to I and T areJ I,i (0) = ∂I(W(µ k • exp(v), x i)) rule, we can decompose these Jacobians asJ I,i (0) = M I M W M L and J T,i (0) = M T M W M L . (C.3)1. M I (resp. M T ) is a 1 × 3 matrix corresponding to the gradient of the warped image I (resp. T ) with parameters µ k (resp. 0):

Table 1 :

 1 Summary of methods fitness to several typical situations of alignement. Noiseless or low noise : σ I , σ T are negligible. Strong noise : σ I or/and σ T are high values. Symmetric noise : σ

		Term of	Term of	Recommended
		structure	noise	approach
		Aαv *	Bαv *
	Noiseless or low noise	negligible	negligible	FCL, ICL,
	with close initialization			ESM, MVACL
	Noiseless or low noise	X	negligible	ESM
	with far initialization
	Symmetric noise	X	X	MVACL, ESM
	Near initialization with	negligible	X	MVACL
	strong asymmetric noise
	Far initialization with	X	X	proposed heuristics
	strong asymmetric noise			AACL and GACL

I ≈ σ T . Near initialization : the motion to estimate ||v * || is close to 0. Far initialization : the motion to estimate ||v * || is large.

Table 3 :

 3 Unified presentation of the algorithm scheme of the ACL approaches.

Table 4 :

 4 Computational time (in ms) of the different approaches. Three steps are distinguished : the precomputation for the template T (P-CC), the extra computation performed by the fast new approaches each time a new image I has to be processed (E-CC), and the per iteration cost performed by each algorithm (I-CC).

	Methods	P-CC	E-CC	I-CC
	FCL	4.22	0	7.00
	ICL	7.60	0	4.53
	ESM	6.73	0	7.16
	AACL ESM	6.38	0	9.08
	F -AACL ESM	6.40	1.33	7.15
	GACL	7.57	0	8.99
	F -GACL	7.61	1.25	7.15

table 5 .

 5

	Conditions		Exp.0			Exp.1			Exp.2			Exp.3			Exp.4			Exp.5
		σp = 6 pixels	σp = 6 pixels	σp = 6 pixels	σp = 12 pixels	σp = 12 pixels	σp = 12 pixels
		SNR=15 dB	SNR=10 dB		SNR=5 dB	SNR=15 dB	SNR=10 dB	SNR=5 dB
	β	0	0.2	0.5	0	0.2	0.5	0	0.2	0.5	0	0.2	0.5	0	0.2	0.5	0	0.2	0.5
	FCL	83.6	85.7	88.2	53.6	62.7	75.0	13.6	18.2	30.6	32.8	37.3	43.2	10.2	13.6	23.3	0.8	1.6	4.8
	ICL	92.4	91.2	88.9	92.4	87.4	75.7	90.4	63.6	32.0	54.4	49.9	42.8	53.6	40.2	24.4	52.7	18.2	4.5
	ESM	95.2	95.0	95.4	91.0	91.3	91.7	59.4	63.9	67.3	67.4	67.2	67.4	52.2	52.4	52.3	18.6	19.7	20.7
	MVACL	92.4	94.0	95.4	92.4	92.2	91.7	90.4	76.0	67.3	54.4	63.4	67.4	53.6	54.2	52.3	52.7	29.0	20.7
	GACL	95.1	95.0	95.1	94.4	92.5	91.4	90.5	73.3	67.6	68.5	67.4	66.4	63.6	56.0	52.2	55.4	27.5	20.5
	AACL FCL	94.5	94.3	94.6	90.7	89.7	90.2	51.6	49.9	58.2	65.1	64.4	64.5	54.4	51.2	50.1	20.3	17.1	17.4
	AACL ICL	94.4	94.2	94.3	93.7	92.4	90.0	91.6	74.3	58.1	65.3	65.1	64.0	61.5	54.8	50.3	56.8	28.6	18.0
	AACL ESM	95.3	95.1	95.2	94.4	92.9	91.6	86.4	73.3	67.8	68.9	68.0	67.1	63.9	56.6	52.6	51.5	28.0	20.6
	F -GACL	95.0	94.8	95.0	94.1	92.4	90.6	89.1	73.0	63.6	67.4	66.4	65.1	61.8	53.7	48.7	50.0	24.0	17.0
	F -AACL ESM	95.2	94.8	95.1	94.0	92.5	90.6	86.3	74.4	64.2	68.0	67.0	66.1	62.6	54.6	50.1	44.7	26.7	18.6

Table 5 :

 5 Average frequency of convergence with respect to noise for the different approaches for characteristic β values. Best overall method is in bold, best methods in each category are underlined.

The benefits of the new methods were studied in the case of image alignment with L2 pixelwise errors. As done by [START_REF] Brooks | Generalizing inverse compositional and ESM image alignment[END_REF] for the inverse compositional and the ESM methods, the proposed approach could be extended to more generic error metrics and motion models to fit particular needs in various applications. We think their properties could be useful in applications where a slighty noisy template would be aligned to a highly noisy image under low-light condition [START_REF] Alter | An intensity similarity measure in low-light conditions[END_REF].

In this paper, we focused on the problem of noise related asymmetry. In applications such as super-resolution [START_REF] Dijk | Superresolution reconstruction for moving point target detection[END_REF], where the images to align can have different resolutions, the derivatives of the low-resolution frame is less reliable than the derivatives of the high resolution estimate. This problem of resolution asymmetry has been thoroughly discussed in [START_REF] Dedeoglu | The asymmetry of image registration and its application to face tracking[END_REF] in term of bias on the estimation of the warp. However, the problem of the reliability of the image derivatives has not been tackled. We think that it is an interesting perspective for future developments, to merge those two sources of asymmetry in a single formulation.

Aknowledgement

The authors would like to thank M. Grossetete of Thales Avionics corp.

for providing the real low light images used in the experimental section.

Appendix A. Second-order approximation of the ACL objective function

In the following we present the main properties used for the derivation of the second-order approximation of the ACL objective function (cf. eq. ( 21)) under noise-free conditions.

Lemma (Second-order linearization of the ACL objective function).

Proof. By developping (8) to second-order around v = 0, we have at each

Proof. We have:

Using lemma A.5 into the previous equation to replace J T nf (-αv * )v * yields (A.7). The special cases are obtained for α = 0 and α = 1.

Note that we generalize here the proof of (A.8) proposed by [START_REF] Benhimane | Homography-based 2d visual tracking and servoing[END_REF] to the generic ACL jacobian.

Lemma (Linear approximation of the image jacobian).

For γ ∈ [0, 1]

Proof. By expanding J I,i (γv) to first-order around 0 we have:

Lemma (Linear approximation of the α-ACL jacobian).

Appendix B. Parametrization of homography

In this section we present the homography parameterization of motion used in the experimental section.

A 2D Homography is a transformation in the projective image plane P 2 . A homography can be represented by a 3 × 3 matrix H :

with µ = [µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 µ 9 ] T in P, which is isomorphic to SL(3), the Special Linear Group of dimension 3 [START_REF] Benhimane | Homography-based 2d visual tracking and servoing[END_REF] satisfying det(H) = 1.

A group action W, fulfilling equations ( 4), ( 5) and ( 6) can be defined from SL(3) to P 2 . For H ∈ SL(3) and x = [x y 1] T ∈ P, we have:

Lie Algebra local reparameterization. For H around identity, H can be reparameterized using the matrix exponential map: