Junfeng Peng

Xuguang Wang

Lisa Denninger

Comparing the preferred driving postures between Chinese and French drivers

Keywords: Vehicle packaging, Driving posture, Anthropometry, Digital human modeling

The effects of seat height and anthropometric dimensions on preferred driving posture were investigated using a multi-adjustable mock-up in the present study. More specifically, Chinese and French drivers were compared. Sixty-one subjects (40 'French-born' and 21 'Chinese-born') participated in the experiment, covering a large range of stature and sitting height to stature ratio. Preferred driving postures were collected from three seat heights fixed at 250, 300 and 350 mm. Apart from hip fore-aft position, seat height only affected trunk-thigh angle while all other inter-segment angles including trunk inclination were kept invariant. Differences were only found in trunk related angles between different participant groups. Shorter persons tended to have a less inclined trunk while they had almost the same trunk-thigh angle as taller persons. Chinese participants had a more opened trunk-thigh angle and a more inclined trunk due to the fact that their hips were positioned more forward in the seat. Results support the assumption that the inter-segmental (joint) angles of the preferred driving posture are not dependent on driver's anthropometry.

Practitioner Summary: Effects of seat height and anthropometric dimensions on preferred driving posture were investigated. The findings would be useful for improving vehicle interior design for a better accommodation of target driver population with a large variation of anthropometric dimensions due to market globalization.

Introduction

Being able to accommodate a target population with a high variety of anthropometric dimensions and individual preferences is one of ergonomic goals in vehicle occupant packaging. In the past, much work has been done to determine the preferred driving postures. [START_REF] Rebiffé | Le siège du conducteur :son adaptation aux exigences fonctionnelles et anthropométriques[END_REF] proposed comfortable joint angle ranges in the sagittal plane for driving from a theoretical analysis of driving activities. Many experimental efforts (Ribouchon 1991, Seidle 1994[START_REF] Porter | Exploring the optimum posture for driver comfort[END_REF][START_REF] Park | Comfortable driving posture for Koreans[END_REF][START_REF] Hanson | Preferred car driving posture using 3-D information[END_REF], Kyung et Nussbaum 2009) were also made to observe the preferred driving postures using a multi-adjustable driving mock-up or real vehicles. Among these studies, contradictory findings on effects of driver's anthropometric dimensions were reported. Park and his colleagues (1999) found trunk-thigh angle was correlated positively with stature, while in another publication [START_REF] Park | Comfortable driving posture for Koreans[END_REF], a negative correlation was found. [START_REF] Porter | Exploring the optimum posture for driver comfort[END_REF] observed a more 'open' posture for tall subjects with bigger elbow and trunk-thigh angle than shorter persons. But they explained that this could be due to the limited adjustment ranges of the rig. Kyung and Nussbaum (2009) found significant stature effect on the left ankle, left hip and neck angle. Ribouchon (1991) observed that the preferred posture was not dependent on stature. This was confirmed by a more recent study by [START_REF] Hanson | Preferred car driving posture using 3-D information[END_REF]. Meanwhile, most of the previous studies were carried out on Europeans or North Americans. There are very few studies with Asian population, especially Chinese drivers. Anthropometrical characteristics of Chinese adults are different from Europeans not only in stature distribution, but also in body segment proportion. For example, according to the anthropometric survey of Chinese adults in 1988, Chinese males had an average stature of 1678 mm against 1719 mm for French male drivers according to the survey by [START_REF] Rebiffé | Enquête anthropométrique sur les conducteurs français[END_REF], while their average sitting heights were respectively 908 and 911 mm for Chinese and French. Among very few studies involving Asian subjects, [START_REF] Park | Comfortable driving posture for Koreans[END_REF] observed that Korean drivers had greater trunk-thigh and knee angles than those recommended in the literature. Kyung (2008) collected preferred driving postures and subjective evaluations from 11 North Americans and 11 Koreans using an adjustable driving mockup. A significant difference in discomfort rating was observed between the two groups. However preferred postures were not compared.

Therefore, there is a need to further investigate the effects of anthropometric dimensions and vehicle interior geometry variables on driving posture. In this paper, the effects of seat height and anthropometric dimensions on preferred driving posture were investigated. More specifically, French and Chinese drivers were compared.

Materials and methods

Participants

Sixty-one drivers participated in the experiment. They all had a driving license for more than one year and drove regularly. They did not suffer any neurological or musculoskeletal disorders. The subjects are divided into three groups according to stature: short (<1654 mm), average (1670-1760 mm) and tall (>1775 mm). In order to have a large variation of sitting height/stature ratio, an effort was made to constitute two comparable groups according to ethnic origin: Chinese and French. Due to recruitment difficulty, most of Chinese-born subjects were in the short and average groups. The main characteristics of the participants according to stature and ethnic origin are described in Table 1. As no Chinese participant was higher than 1805 mm, a sub-sample was constituted by excluding those taller than 1805 mm. The experimental protocol was approved by the ethical committee of IFSTTAR (French Institute of Science and Technology for Transport, Development and Networks). Informed consent was given before participating in the experiment.

Experimental facilities

A multi-adjustable vehicle mockup shown in Figure 1, composed of an accelerator pedal, a clutch pedal, a foot rest, a steering wheel and a seat, was used in this study. The available adjustments are listed in Table 2. Most of them were motorized and could be controlled by subjects themselves easily through an intuitive touch-screen, while others were controlled manually by an experimenter. The accelerator pedal height being also adjustable, the ball of foot reference point (BOFRP) defined in SAE J4004 was thus not fixed. The geometrical accelerator pedal center (APC) was used as the origin for the longitudinal x-axis. The vertical zaxis was defined from the floor. A Peugeot 508 seat, considered as a high-end product with electric adjustments in fore-aft position, height, lumbar support, cushion angle and back inclination, was used. Its cushion length was also adjustable from 364 to 399 mm, measured horizontally from the seat front edge to the H-point. The seat track was mounted on a platform adjustable both in height and inclination, providing an adjustable seat height from about 150 mm to 500 mm. An extended range of seat adjustment in x was ensured by changing the accelerator pedal in addition to the adjustment range of 240 mm of seat track itself. Seat cushion angle could be adjusted by the use of its own tilt mechanism with a range up to 5°. If a larger rotation was required, the platform on which mounted the seat track could also be rotated. Large seat back adjustment range was also provided from the vertical to almost the horizontal. The steering wheel was mounted on a separate pillar, providing a large adjustment range in both x and z directions. The clutch pedal could be adjusted independently with respect to the accelerator pedal. Travel angle of clutch pedal could be chosen by subjects from the five predefined values: 9°, 16°, 23°, 30° and 37°. Travel length was fixed at 150 mm. The pedal resistance at the end of depression was not explicitly controlled as in our earlier studies (Wang et al, 2000, Pannetier and[START_REF] Pannetier | A comparison of clutching movements of freely adjusted and imposed pedal configurations for identifying discomfort assessment criteria[END_REF]. The fore-aft position the foot rest could also be changed as well as its inclination. There were five discontinuous foot rest inclinations: 29°, 36°, 43°, 51° and 60°. No brake pedal was mounted.

Reflective markers were attached to the body as well as to all moveable parts of the mock-up. A VICON optoelectronic system with 14 MX40 cameras was used for capturing the positions of these markers. Manual by experimenter

Experimental conditions and procedure

Three seat heights at 250, 300 and 350 mm were tested in this study, covering a large range of passenger vehicles. For each seat height, subjects were asked to find their preferred driving vehicle interior dimensions by changing all available adjustments with the fore-aft position of the accelerator being fixed. The configuration with seat height of 300 mm was repeated three times. The trial order was randomized.

Prior to the experiment, a set of preliminary work had to be carried out. After the arrival of the subject in the motion capture room, the purpose of the experiment was first explained. Then, the participant was asked to change into a gym-suit and 32 anthropometric measurements were collected. Reflective markers were attached to the body. The participants were photographed in both a standing and a sitting posture in a calibrated space. These postures were also captured with the motion capture system for the kinematic motion reconstruction. For each test condition, initial positions of interior vehicle elements were set to their extreme positions so that participants had to use all available adjustments. An adjustment sequence was suggested but not mandatory: accelerator pedal (only vertical position), seat (fore-aft position, cushion angle, back angle), footrest (for-aft position, inclination), clutch pedal (fore-aft position, vertical position, travel inclination), steering wheel (vertical and fore-aft position, inclination). The participants were free to go back to the previous adjustments at any moment, and to take as long time as needed. Once the subjects found their preferred driving positions, a full movement of depressing clutch pedal was recorded by the Vicon motion caption system. The participants were instructed to place their hands on the steering wheel at the 10-to-2 o'clock position and to look forward as they were driving. They were asked to put their left foot on the foot rest, to fully depress the clutch pedal to its end, and then to move the left foot back on the foot rest. The right foot was asked to put on the accelerator without depressing. Subjects were required to leave the mockup after each trial and the adjustments were reset for a new test condition.

Data processing and analysis

All captured motions were reconstructed using the RAMSIS human model by minimizing the distance between the captured and model-based markers positions (see Ausejo and Wang 2008, for the principle of motion reconstruction). A set of postural parameters, defined Table 3 and illustrated in Figure 2, were calculated from reconstructed joint centers for describing the driving postures at the rest, beginning and end of clutch pedal depression. In this paper, only the parameters of the trunk and the right side limbs are reported at the rest position, for which the left foot was on the foot rest and the right foot on the accelerator pedal without depressing. Two-way analysis of variance ( ANOVA) was mainly used to assess the effects of participant group, either by stature (SG or by ethnic group (OG) and seat height (SH). P-value was fixed at 0.05 for statistical tests.

Results

The means and standard deviations of the postural variables by seat height and subject group were given in Error! Reference source not found.. The effects of seat height and participant group by stature or by ethnic origin are also indicated. Neither seat height nor participant group had a significant effect on shoulder, elbow and knee angles. Seat height only affected trunk-thigh (hip) angle. Higher the seat was, larger the trunk-thigh angle was. Interestingly, trunk inclination with respect to the vertical was not affected by seat height. Significant differences between three stature groups were found in trunk inclination but not in hip angles. Short subjects had a less inclined trunk by about 3° than tall volunteers. In addition to seat foreaft position and trunk inclination, signification difference in hip angle was observed between the two ethnic groups. Chinese subjects had a more inclined trunk by 2.6° and a larger trunk-thigh angle by 3.8° than French group on average.

Discussion

In this paper, the effects of seat height and anthropometric dimensions on preferred driving posture were investigated. More specifically, French and Chinese drivers were compared. The main findings can be summarized as follows:

• Seat height only affected trunk-thigh angle while all other inter-segment angles including trunk inclination were kept invariant. • Differences were only found in trunk related angles between different participant groups. Shorter persons tended to have a less inclined trunk while they had almost the same trunk-thigh angle as tall persons. Chinese participants had a more opened trunk-thigh angle and a more inclined trunk.

Results show that trunk inclination was unaffected by seat height. This is in agreement with the observations by [START_REF] Reed | Effects of Vehicle Interior Geometry and Anthropometric Variables on Automobile Driving Posture[END_REF]. This is probably due to the fact that the head has to be kept in a same orientation whatever seat height for a good forward vision required for driving as analyzed by [START_REF] Rebiffé | Le siège du conducteur :son adaptation aux exigences fonctionnelles et anthropométriques[END_REF]. As knee angle and trunk inclination were invariant to seat height change, a higher seat leads to a more opened trunk-thigh angle for a same person.

Contrary to the observations by [START_REF] Porter | Exploring the optimum posture for driver comfort[END_REF] who reported that taller drivers tended to adopt a more opened posture with a larger trunk, knee, shoulder and elbow angle, no significant differences were observed between three stature groups in all inter-segment angles in the present study, in agreement with observations by [START_REF] Hanson | Preferred car driving posture using 3-D information[END_REF]. Instead, only a significant difference in trunk inclination with respect to the vertical direction between three stature groups was observed. This could be explained, at least partially, by the invariance of inter-segmental angles. If a short and a tall person adopt a same posture (with the same joint angles) for a same seat height, the shorter person has to rotate forward around the pelvis in order to reach the accelerator pedal, leading to a less inclined trunk.

One of main objectives was to compare the preferred driving postures between Chinese and French drivers. It was observed that Chinese participants had a more opened trunk-thigh angle and a more inclined trunk. A further investigation of the offset between hip joint center and seat H point showed that Chinese participants tended to sit in a more forward position in the seat. Figure 3 compares the offsets in x between hip joint center and seat H-point between different subject groups. An effect of seat height can be clearly observed for short group. Chinese group had a bigger offset than French group. This might be explained by different body proportions between two ethnic groups. As showed by the anthropometric surveys in the past (GB1988 and [START_REF] Rebiffé | Enquête anthropométrique sur les conducteurs français[END_REF], Chinese has a larger sitting height/stature ratio than French. In the present study, sitting height/stature ratio for the Chinese participants were significantly larger than French group (Table 1). In other words, for a same stature, Chinese has shorter lower limbs than French. The seat used in the present study had a minimum cushion length of 364 mm, measured horizontally from the seat front edge to the H-point. [START_REF] Reed | Survey of auto seat design recommendations for improved comfort[END_REF] recommended that the cushion length, measured along the thigh line with an angle of 15°, should not exceed 305 mm from the seat H-point to seat front edge based on U.S. population data. This is equivalent to a horizontal distance of 316 mm from the seat front edge to the H-point. Compared with this recommendation, the seat used in this study had a long seat cushion. This suggests that the seat designed for Chinese market should be different than that used in French market. In the future, the effects of seat on preferred posture need to be investigated.

Apart from the limitation that only one seat was used, one important limitation is that the experiment was carried out in the laboratory without considering the visual constraints. In order to investigate the body segment proportion effect on driving posture, a big effort was made to recruit equal number of Chinese and French subjects in each stature groups (short, average and tall). However, nearly no tall Chinese were recruited because the experiments were carried out in France. It would be interesting to investigate the differences between Chinese and European drivers using a larger sample.

Conclusion

Apart from trunk-thigh angle, inter-segmental angles and trunk inclination are largely unaffected by seat height. Our results support the assumption that the inter-segmental (joint) angles of the preferred driving posture are not dependent on anthropometric dimensions. With a same set of joint angles, different hip and eye positions are reached for differently sized people for a given vehicle configuration. Due to difference in body proportion, Chinese tended to sit more forward in a seat than French. Further investigation on the effects of seat with a larger sample is required in the future.

Figure 1 .

 1 Figure 1. Experimental vehicle mock-up Table 2. List of adjustable vehicle interior parameters Adjustable Parameter Type of control Seat H-point vertical position to floor Touch-screen Cushion angle defined according to SAE J826 (MAR2008) Control button or touch-screen Seat H-point horizontal position aft of APC Control button Back angle defined according to SAE J826 (MAR2008) Control button Vertical position of steering wheel center to floor Touch-screen Steering wheel horizontal position aft to APCx Manual by subject Steering wheel angle Manual by subject Accelerator pedal height to floor Touch-screen Clutch pedal undepressed horizontal position aft of APC Touch-screen Clutch pedal height to floor Touch-screen Clutch pedal travel angle Manual by experimenter Footrest reference horizontal point aft to APC Touch-screen Footrest angle Manual by experimenter

Table 3 .

 3 Figure 2. Definition of postural variables

Figure 3

 3 Figure 3 Offsets between hip joint centre and seat H-point by stature group (a) and by ethnic group (b) in three seat heights. Negative value means that the hip is more forward than H-point. Only participants with stature less than 1805 mm were considered when comparing two ethnic groups.

Table 1 .

 1 Main characteristics of participants by stature and ethnic origin. SH, AL and S stand for sitting height, arm length as the sum of the upper arm length and the forearm length, and stature

	Group	M/F	Stature (S) (mm)		Sitting height (SH) (mm)	SH/S (10 -2 )	AL/S (10 -2 )	Weight (kg)
			Mean±SD	Max/min
	Short	1/19	1582±41	1645/1490	859±23	905/815	54.3±1.2	43.1±1.4	54.4±7.5
	Average	14/6	1715±32	1760/1670	915±29	970/860	53.4±1.5	43.4±1.4	67.0±8.8
	Tall	21/0	1846±62	2000/1775	960±26	1000/910	52.0±1.3	43.8±1.6	74.3±11.2
	All	36/25	1717±119	2000/1490	912±49	1000/815	53.2±1.6	43.5±1.5	65.4±12.4
	French	12/15	1683±88	1800/1520	897±46	975/815	53.3/1.3	43.6±1.5	63.1±9.4
	Chinese	11/10	1657±86	1805/1490	899±40	980/840	54.3/1.1	42.6±1.0	59.8±10.6


Loïc Brevault

Mathieu Balesdent

Sebastien Defoort

Preliminary study on launch vehicle design -applications of Multidisciplinary Design Optimization methodologies

The design of complex systems such as launch vehicles involves different fields of expertise that are interconnected. To perform multidisciplinary studies, concurrent engineering aims at providing a collaborative environment which often relies on data set exchange. In order to efficiently achieve system level analyses (uncertainty propagation, sensitivity analysis, optimization, etc.) it is necessary to go beyond data set exchange which limits the capabilities of performance assessments. Multidisciplinary Design Optimization (MDO) methodologies is a collection of engineering methodologies to optimize systems modeled as a set of coupled disciplinary analyses and is a key enabler to extend concurrent engineering capabilities. This paper is focused on several examples of recent developments of MDO methodologies (e.g. MDO with transversal decomposition of the design process, MDO under uncertainty) with applications to launch vehicle design to illustrate the benefices of taking into account the coupling effects between the different physics all along the design process. These methods enable to manage the complexity of the involved physical phenomena and their interactions in order to generate innovative concepts such as reusable launch vehicles beyond existing solutions.

INTRODUCTION

The design of complex systems such as launch vehicles, satellites, aircraft, etc. involves different fields (disciplines) of expertise (e.g. aerodynamics, structure, trajectory) that interact. To design new aerospace vehicles, one of the challenges is to manage the complexity of the involved physical phenomena and their interactions in order to generate innovative concepts beyond existing solutions, with a generic methodological approach to be applied to very different systems such as launchers or aircraft. Concurrent engineering aims at providing a collaborative and simultaneous engineering working environment in order to perform multidisciplinary studies to design complex systems. The concurrent engineering approach relies on a process, a multidisciplinary team, an integrated design model, a facility and a software infrastructure [START_REF] Xu | Overview on the development of concurrent design facility}[END_REF]. The existing concurrent engineering process (developed for instance at the Concurrent design Facility (CdF) of the European Space Agency, at the Centre d'Ingénierie Concourante of the Centre National d'Etudes Spatiales (CNES), in the space industry such as Airbus Safran Launcher) relies on an iterative process which involves engineering model interfaces (often through Microsoft Excel spreadsheets) handled by the field experts and a common data set used as inputs for the engineering models (Figure 1) [START_REF] Xu | Overview on the development of concurrent design facility}[END_REF]. At each iteration of the concurrent engineering design process, the data set of each field is updated with the results from the discipline models and collaborative work sessions are organized. This approach offers advantages in terms of collaborative work and is a first approach in terms of interaction management between the disciplines. However, this classical concurrent engineering approach has several limitations:  Use of important design margins in order to ensure multidisciplinary consistency between the disciplines,  Difficulty to implement an optimization process at the system level. In classical concurrent engineering approach, the system design results from the concatenation of local disciplinary optimizations (Figure 1). However, due to possible antagonistic discipline objectives, a difficult search for compromise between these conflicting tasks needs to be performed.  Difficulty to implement system level analyses (uncertainty propagation, reliability assessment, sensitivity analysis, optimization, etc.) in an automated process which could offer more information to the decision makers earlier in the design process.

In order to overcome the existing concurrent engineering limitations, a paradigm shift in terms of design process and analysis tools is necessary and Multidisciplinary Design Optimization (MDO) methodologies could be a key enabler to extend concurrent engineering capabilities. MDO is a collection of engineering methodologies to optimize systems modeled as a set of coupled disciplinary analyses. For instance, launch vehicles are customarily decomposed into interacting submodels for aerodynamics, trajectory, propulsion, mass and structure. The design of launch vehicle consists in determining its architecture to achieve a given mission and this launcher architecture is usually found through an optimization problem solving. Taking into account the different disciplines requires modelling and managing the interactions between them all along the optimization process.

Different topics related to MDO methodologies to facilitate aerospace vehicle design have been investigated: a. Deterministic classical MDO formulations, b. Deterministic dedicated MDO formulations relying on transversal decomposition of the design process, c. MDO formulations under uncertainty in order to ensure the reliability and robustness with respect to uncertainty. This paper is focused on several examples of recent use of MDO techniques at ONERA in order to illustrate the progress in terms of MDO methodologies. In the rest of the paper, these three topics will be explored relying on illustrative example on launch vehicle design problem. In section 2, both classical and dedicated deterministic MDO methods for launch vehicle design are explored. In section 3, MDO methodologies in the presence of uncertainty are presented in order to ensure the reliability and robustness of the launch vehicles. In section 4, perspectives to use MDO methodologies for concurrent engineering sessions are discussed.

DETERMINISTIC MDO METHODOLOGIES FOR LAUNCH VEHICLE DESIGN a. Classical MDO formulations

The design space exploration to design a launch vehicle requires formalizing the optimization problem to solve, meaning from the system and mission requirements, to identify the disciplines involved, the interactions and couplings between the disciplines, the design variables, the objective function, the constraints, etc. The aim is to build a design process that is compliant with certain requirements: robustness of the performance, computational cost of the process, dimension of the design space, or the constraints to be fulfilled. A major difference with the classical concurrent engineering approach is that in addition to share a common data set, the disciplinary models are chained offering possibility to account for interdisciplinary couplings and to ensure consistency between the discipline executions. Several organizations of the optimization process have been proposed, called MDO formulations (Figure 2). These formulations may be classified according to four categories (Balesdent et al., 2012a):

 Coupled formulations: easiest approaches, ensure interdisciplinary coupling consistency at each iteration of the optimization process (e.g. Multi Discipline Feasible (R. J. [START_REF] Balling | Optimization of Coupled Systems -A Critical overview of Approaches[END_REF]),  Decoupled formulations: enable parallel discipline evaluations and ensure interdisciplinary coupling consistency at the convergence of the optimization problem (e.g. Individual Discipline Feasible [START_REF] Balling | Optimization of Coupled Systems -A Critical overview of Approaches[END_REF]),  Single-level formulations: the optimization process is driven by a single optimizer at the system-level (e.g. All At Once [START_REF] Balling | Optimization of Coupled Systems -A Critical overview of Approaches[END_REF] ),  Multi-level formulations: the optimization process is driven by one optimizer at the systemlevel associated with several optimizers at the sub-system level (discipline level) in order to ease the system level optimizer convergence (e.g. Collaborative Optimization [START_REF] Braun | Use of the Collaborative Optimization Architecture for Launch Vehicle Design, 6thAIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Part I[END_REF], Analytical Target Cascading [START_REF] Allison | On the Use of Analytical Target cascading and Collaborative Optimization for Complex System Design, International Society for Structural and Multidisciplinary Optimization[END_REF]).

Figure 2: Classical general MDO formulations in the literature

These formulations have the advantage of being generic and adaptable to any complex system design. In order to illustrate how these global design issues could be taken into account to provide decision-makers a full understanding of the choices at stake, a system study was set up at ONERA, using internal facilities and design methods. This study explores the impact of reusability onto a global launcher design. The starting point of this study is threefold:  First, the baseline architecture of the current Ariane 6 design is considered as a starting point.

The study logic is to try to derive a reusable launcher (with 1 st stage recovery) from this configuration, with minimal architecture and technology changes.  Second, the 1 st stage recovery is assumed to be performed by a "space X-like" vertical landing on an infrastructure located in the ocean, and using only rocket propulsion during re-entry.  Third, the main design assumption will be the use of a 80-ton-thrust LOX/CH4 engine, which has been stated as a promising solution. These assumptions are not established to be the optimal ones, but provide a common reference to study several launcher variants (Figure 3). From this starting point, the two stage launcher is fully redesigned to meet a delivery capability of 5 tons on a Geostationary Transfer Orbit (250km x 36000km) with a minimal Gross Lift-Off Weight (GLOW). Several different configurations have been explored and optimized (Figure 3). A fastresponse sizing loop relying on a Multi Discipline Feasible (MDF) formulation was used to jointly optimize the following design variables:  the launcher geometry (stage diameters, volume of propellants),  the propulsion system (chamber pressure, nozzle expansion ratio, mixture ratio)  the 3-phase trajectory (two-stage ascent with possible ballistic phases, controlled by pitch angle management, and descent, controlled by a pitch angle law and the duration and intensity of propulsion boosts; and reentry). The disciplines involved are the propulsion, the mass budget and geometry, the aerodynamics and the trajectory. The discipline models are provided by field experts and interdisciplinary couplings between the disciplines have been identified in order to enable code chaining and to account for representative interdisciplinary interactions.

A MultiDisciplinary Analysis (MDA) based on iterative fixed point method is used to satisfy the interdisciplinary couplings at each iteration of the system level optimizer, ensuring the physical relevance of the designed launch vehicle. MDF is used to find the optimal configuration of the launcher, satisfying the design and interdisciplinary coupling constraints.

Figure 4: Ascent velocity and descent trajectory for different concepts

After MDF runs, analyses with field experts and system engineers is performed in order to understand the design choices and to analyse the interdisciplinary interactions and their aftermaths on the design process. As a main outcome of this study, it appears that providing the 1 st stage with a recovery capability induces a full redesign of the launcher (i.e. propulsion choices, both from the propellant and the engine points of view, staging, trajectory). A trade-off is observed between:

 the staging Mach number (the higher, the easier the design of the upper stage, but the greater the re-entry difficulties),  the propellant mass used for 1 st stage recovery (the lower, the better for global performance, but the higher the re-entry loads and thus the structural impacts),  the design peak dynamic pressure during re-entry (the higher, the lower the fuel consumption, but the harder the loads on the re-used stage). The order of magnitude of GLOW penalty to provide the launcher with reusability seems to be around 25%, which is to be translated into cost increase for the same payload, and compared to the gain of reusability. Nevertheless, it seems that an easy derivation of a reusable version from current Ariane 6 concepts is not straightforward, and that maximum reuse of existing stages (boosters or upper stage) or even existing engines (Vinci or Vulcain) is doubtful or should be consolidated.

From the design point of view, due to the antagonistic discipline objectives (for instance ascent and descent trajectories), classical concurrent engineering approach does not offer the possibility to find global optimal solution. Moreover, designing the entire vehicle to provide new capability (here reusability of the 1 st stage) impacts all the disciplines through the couplings and the technological choices. Therefore, using a MDO approach enables to take into account the coupling effects between the different physics all along the design process and analysis which is a very important aspect for such complex system design. To quantify their performance and reliability tightly coupled multidisciplinary has been used in order to provide key decision arguments to the stakeholders In conclusion, MDO approach offers key advantages compared to concurrent engineering methods such as the possibility to explore large variety of configurations in parallel (nine concepts in this example) to assess the best architecture relatively to mission requirements.

b. SWORD: Stage-Wise decomposition for Optimal Rocket Design

The classical MDO formulations introduced in the previous section are flexible enough to be adapted to the design of any complex system such as launch vehicle. However, launch vehicle design present particularities, notably the importance of the trajectory discipline compared to the other disciplines. Exploiting these specificities in a dedicated MDO formulation might improve the launch vehicle design process. Specific formulations for launch vehicle design have been proposed such as the Stage-Wise decomposition for Optimal Rocket Design (SWORD) (Balesdent et al., 2012b). The classical MDO approaches as illustrated in the previous paragraph, decompose the design process according to the involved disciplines (propulsion, aerodynamics, sizing, trajectory, etc.).

In order to lay the trajectory optimization at the center of the MDO process, SWORD splits up the launch vehicle design problem according to the different flight phases and transforms the global MDO problem into the coordination of smaller ones. Each stage is optimized separately and the different stages are coordinated through the trajectory optimization, via the state vectors at the stage separations. In the SWORD formulations, the subsystems are not the disciplines but instead the different launch vehicle stage optimizations incorporating all the required disciplines to the stage design. Four different formulations have been proposed depending on the decomposition process and the interdisciplinary coupling constraint handling. One of the SWORD formulations consists of a hierarchical decomposition of the design process according to the launch vehicle stages (Figure 5). In this formulation, the objective function is assumed to be decomposed according to the different stages. In practice, the GLOW is often minimized in launch vehicle design process (Balesdent et al., 2012a) and it can be decomposed as the sum of the stage masses plus the fairing and payload masses. The different stage are coupled, a change in the upper stage mass would introduce a modification of the lower stage masses in order to ensure mission success. Also, the cost of launch vehicle follows the same decomposition (i.e. sum of the stage cost).

Figure 5: Example of SWORD formulation

In order to assess the efficiency of SWORD, a comparison with MDF formulation has been performed on a launch vehicle design problem using the same classical Genetic Algorithm in a very large search space. The problem to be solved is the optimization of a three-stage cryogenic launch vehicle including 40 design variables (trajectory command law, propellant masses, stage diameters, propulsion chamber pressures, etc.) [START_REF] Balesdent | Multidisciplinary Design Optimization of launch vehicles[END_REF]. The objective function to be minimized is the GLOW. The payload mass is equal to 4 tons and the target orbit is a GTO. Three comparative criteria for the formulations have been selected:

• The best found design at the stopping time of the optimization algorithm;

• The time elapsed to find a first feasible design from random initialization;

• The improvement of the objective function during the optimization process. SWORD clearly outperforms MDF in the case of global search optimization in a very large search space (Figure 6). SWORD allows to significantly improve the global search efficiency with respect to the MDF formulation by reducing in average by 4 the required computation time to find a first feasible design, finds a lighter design than MDF and allows to efficiently improve the current design all along the optimization process whereas MDF presents some difficulties to improve the objective function. SWORD is interesting as it ensures the consistency of the mass coupling during the optimization process and does not require equality constraints at the system level.

MDO METHODOLOGIES UNDER UNCERTAINTY FOR LAUNCH VEHICLE DESIGN

The early design phases of LV are often characterized by the use of low fidelity analyses as well as by the lack of knowledge about the future system design and performance. Indeed, the low fidelity analyses are employed due to the non-possibility to build high fidelity models for each possible architecture to explore the entire design space. This global exploration results in repeated discipline evaluations which are impossible to perform at an affordable computational cost with high fidelity models. Moreover, to increase the performance of the launch vehicles and to decrease their costs, space agencies and industries introduce new technologies (new propellant mixture such as liquid oxygen and methane, reusable rocket engines) and new architectures (reusable first stage for launch vehicles) which present a high level of uncertainty in the early design phases. Incorporating uncertainties in MDO methodologies for aerospace vehicle design has thus become a necessity to offer improvements in terms of reduction of design cycle time, costs and risks, robustness of launch vehicle design to with respect to uncertainty along the development phase, and increasing the system performance while meeting the reliability requirements.

If uncertainties are not taken into account at the early design phases, the detailed design phase might reveal that the optimal design previously found violates specific requirements and constraints. In this case, either the designers go back to the previous design phase to find a set of design alternatives, or they perform design modifications at the detailed design phase that could result in loss of performance. Both options would result in a loss of time and money due to the rerun of complex simulations. Moreover, uncertainties are often treated with safety margins during the design process of launch vehicles which may result in over conservative designs therefore an adequate handling of uncertainty is essential [START_REF] Brevault | Multi-level Hierarchical MDO formulation with functional coupling satisfaction under uncertainty, application to sounding rocket design[END_REF]. Uncertainty-based Multidisciplinary Design Optimization (UMDO) aims at solving MDO problems under uncertainty.

Incorporating uncertainty in MDO methodologies raises a number of challenges which need to be addressed. Being able, in the early design phases, to design a multidisciplinary system taking into account the interactions between the disciplines and to handle the inherent uncertainties is often computationally prohibitive. In order to satisfy the designer requirements, it is necessary to find the system architecture which is optimal in terms of system performance while ensuring the robustness and reliability of the optimal system with respect to uncertainty. One of the key challenges is the handling of interdisciplinary couplings in the presence of uncertainty. Most of the existing UMDO formulations are based on an adaptation of the single-level MDF formulation in the presence of uncertainty [START_REF] Brevault | Multi-level Hierarchical MDO formulation with functional coupling satisfaction under uncertainty, application to sounding rocket design[END_REF]. These methods are very computationally expensive because they combine the computational cost of the optimisation problem solving, the interdisciplinary coupling by using an MDA (i.e. loop between the disciplines) and the uncertainty propagation often performed by Monte Carlo simulations. Alternatives exist (e.g. SORA [START_REF] Du | Sequential optimization and reliability assessment method for efficient probabilistic design[END_REF]) but introduce simplifications that can lead to wrong design.

In order to tackle the computational cost of coupled MDO formulations and preserve the validity of the found design, two new UMDO formulations [START_REF] Brevault | Multi-level Hierarchical MDO formulation with functional coupling satisfaction under uncertainty, application to sounding rocket design[END_REF], Brevault et al. 2016) with interdisciplinary coupling satisfaction for all the realizations of the uncertain variables have been elaborated. With the aim of ensuring multidisciplinary feasibility, a new technique has been proposed based on a parametric surrogate model (Polynomial Chaos Expansion) of the input coupling variables and a new interdisciplinary coupling constraint to guarantee the validity of the of the interdisciplinary coupling satisfaction when impacted by uncertainties. This technique enables the system-level optimizer to control the parameters defining the surrogate model of the input coupling variables in addition to the design variables. Therefore, it enables to decouple the disciplines while ensuring at the UMDO problem convergence that the functional relations between the disciplines are the same as if a coupled approach using MDA had been used. The two proposed formulations rely on this technique to handle interdisciplinary couplings. The first formulation is a single-level approach inspired from Individual Discipline Feasible (IDF) and adapted to the presence of uncertainty. This approach, called Individual Discipline Feasible -Polynomial Chaos Expansion (IDF-PCE) [START_REF] Brevault | Multi-level Hierarchical MDO formulation with functional coupling satisfaction under uncertainty, application to sounding rocket design[END_REF]) (Figure 7), allows to ensure multidisciplinary feasibility for the optimal solution while reorganizing the design process through a decomposition strategy.

Figure 7: IDF-PCE (left) and MHOU (right) formulations

The second formulation is a multi-level approach inspired from SWORD (Balesdent et al., 2012b), which has been modified to take into account uncertainty and to maintain the equivalence with coupled approaches in terms of multidisciplinary feasibility. This formulation (Figure 7), named Multi-level Hierarchical Optimization under Uncertainty (MHOU) [START_REF] Brevault | Multi-level Hierarchical MDO formulation with functional coupling satisfaction under uncertainty, application to sounding rocket design[END_REF], introduces multi-level optimization of the disciplines and is particularly adapted for launch vehicle design.

These formulations have been applied on a two stage launch vehicle design test problem and compared to the MDF under uncertainty formulation. Four coupled disciplines are involved in this problem: the aerodynamics, the mass budget, the propulsion and the trajectory (Figure 8). This test case consists in minimizing the expected value of the GLOW under the constraint of injecting a 5t payload into a GTO orbit at the perigee of 250km. The design problem has 27 design variables (stage diameter, propellant mass, thrust, mixture ratio, derating factor, trajectory control law) and three uncertain variables are taken into account (1 st stage specific impulse uncertainty, 2 nd stage thrust uncertainty and 2 nd stage dry mass uncertainty). The probability of failure to reach the target orbit is estimated by Subset Simulation [START_REF] Au | Estimation of small failure probabilities in high dimensions by subset simulation[END_REF] using Support Vector Machine [START_REF] Dubourg | Metamodel-Based Importance Sampling for Structural Reliability Analysis[END_REF] of the limit state defining the failure. A patternsearch algorithm [START_REF] Audet | Analysis of Generalized Pattern Searches[END_REF] is used for the system level optimizer. As presented in Figure 9 (upper left), the ONERA MDO formulation (the decoupled single-level IDF-PCE) under uncertainty is more efficient (by a factor of 11) in terms of number of calls to the different disciplines than the coupled approach MDF under uncertainty. The importance of taking into account the presence of uncertainty is highlighted in the bottom of Figure 9. Indeed, the same problem has been solved with a deterministic approach (considering all the uncertainties frozen at their expected values, bottom left of Figure 9). An optimal launch vehicle has been found with a corresponding trajectory. Then, the optimal deterministic vehicle has been perturbed by the presence of uncertainties and has resulted in an important injection dispersion of the payload into orbit (bottom middle Figure 9). As illustrated on the bottom right of Figure 9, taking into account the presence of uncertainty directly in the design process enables to design a vehicle that is robust to to uncertainty allowing to ensure the required injection precision for the payload. Using a MDO approach enables to perform in a simple manner uncertainty propagation and optimization while taking into account the potential cascading effects on the different disciplines through the interdisciplinary couplings.

CONCLUSIONS

In order to develop innovative aerospace vehicle concepts it is necessary to achieve increasingly complex system integration studies and one faces the challenge of developing efficient, robust and adapted design methodologies. Comprehensive study of process decomposition, uncertainty quantification, high-fidelity tool integration and formulation of the optimization strategy are mandatory, but this theory must always be tested and validated on 'real-life' design cases.

To enhance collaborative efficiency between system engineers, disciplinary experts, CAD designers and decision makers concurrent engineering should rely on dedicated methodologies for system design studies. As illustrated in Sections 2 and 3 Multidisciplinary Design Optimization tools could enhance concurrent engineering approach by enabling fully integrated multidisciplinary analysis to ensure appropriate interdisciplinary consistency while allowing to assess system tradeoff between antagonist discipline objectives. Concurrent engineering sessions could be improved by setting a multidisciplinary collaborative framework enabling to quickly assess, compare and submit to experts and decision makers the different concepts under evaluation. To do so, a proper description of models, interdisciplinary couplings, data exchange process, objective functions and constraint functions is required in order to appropriately formulate the MDO problem. Moreover, capitalization of disciplinary models and libraries of methodologies are required to develop such a framework. Using MDO methodologies between concurrent engineering sessions would enable to analyze the system performance with the disciplinary experts focusing on the interdisciplinary couplings improving the understanding of the impact of each discipline on the system performance.

From a design capability point of view, the combined use of an integrated MDO approach (for instance for launch vehicle design, the whole interactions between disciplines being modelled, and a joint geometry/trajectory optimization being performed) and a collaborative environment (allowing exchanges between experts and design space visualization and exploration) enables to explore quickly several design options and provide sensitivities. As illustrated in this paper, dedicated MDO formulations for launch vehicle design are more efficient in terms of quality of the found solution and computational cost than classical integrated approach. Moreover, the presented results outlined the importance of taking into account uncertainties in the early design phases in order to incorporate the lack of knowledge and to design robust systems.

Further improvements in MDO methodologies are required to be applied on difficult industrial problems. For instance, MDO for high dimensional problems (e.g. high dimensional couplings such as aerodynamic and structure meshes) is still a major challenge. Another important issue is the appropriate handling of mixed continuous, discrete and categorical variables in MDO problems in order to make technological choices along with system performance optimization. Development of new methodologies is required to tackle these types of problems in order to be applied on complex industrial problems.

ACKNOLEDGMENT

This work has been partially supported by CNES. This support is gratefully acknowledged.

Figure 1 :

 1 Figure 1: Concurrent engineering design process (left) and optimization (right)

Figure 3 :

 3 Figure 3: Resulting optimal configurations for 5-8t mission in GTO with a reusable first stage (blue: LOX/CH4, red: LOX/LH2, black: solid propulsion, named from left to right EA, EB, R1A, R1B, R1C, R2A, R2B, R3A, R3C)

Figure 6 :

 6 Figure 6: Comparison of SWORD and MDF (average over 10 optimizations)

Figure 8 :

 8 Figure 8: MultiDisciplinary Analysis for the launch vehicle design problem under uncertainty


P Trouvé-Peloux

J Sabater

A Bernard-Brunel

F Champagnat

G Le Besnerais

T Avignon

Turning a conventional camera into a 3D camera with an add-on

We propose to add an optical component in front of a conventional camera to improve depth estimation performance of Depth from Defocus (DFD), an approach based on the relation between defocus blur and depth. The add-on overcomes ambiguity and dead zone, which are the fundamental limitations of DFD with a conventional camera, by adding an optical aberration to the whole system which makes the blur unambiguous and measurable for each depth. We look into two optical components: the first one adds astigmatism and the other one chromatic aberration. In both cases we present the principle of the add-on and experimental validations on real prototypes.

INTRODUCTION

The development of 3D applications in domotic, robotic or augmented reality, leads to a growing interest for compact 3D cameras. Most 3D devices are based on stereoscopy, a well know approach for depth estimation based on the measure of disparity between images from different cameras [1,2]. Recent systems also project an infrared pattern on the scene to counteract the potential lack of texture [3]. However, stereoscopy requires two cameras, increasing the size of the system and leading to calibration issues. An interesting alternate option is Depth from Defocus (DFD) [START_REF] Pentland | A new sense for depth of field[END_REF]. DFD exploits the relation between depth and defocus blur, and requires a single lens only, thus opening the way to very compact 3D cameras. While some DFD approaches require multiple images with various optical parameters [START_REF] Subbarao | Parallel depth recovery by changing camera parameters[END_REF][START_REF] Pasinetti | A depth from defocus measurement system using a liquid lens objective for extended depth range[END_REF], we focus here on Single Image DFD (SIDFD) which is more suited to dynamic scenes capture. However, as pointed out in [START_REF] Trouvé-Peloux | Theoretical performance model for single image depth from defocus[END_REF], SIDFD with a conventional lens suffers from depth ambiguity (a similar blur amount can be measured in front or behind the camera in-focus plane), and from a "dead zone" within the Depth of Field (DoF) of the camera where no blur variation can be measured. Specific camera settings allow to mitigate these limitations, e.g. focusing at infinity to avoid blur ambiguity and choosing a large camera aperture do reduce the DoF. However these settings induce a large defocus blur, leading to a significant loss of depth estimation accuracy, as demonstrated in [START_REF] Trouvé-Peloux | Theoretical performance model for single image depth from defocus[END_REF].

In this paper, we propose a cheap and user friendly approach to give a 3D ability to a conventional camera. It consists in mounting an optical component, which we call an "add-on", in front of the lens. This component adds an aberration which reinforces the variability of the defocus blur with respect to depth, thus improving DFD performance. We describe two different components, based on two different optical aberrations: astigmatism and chromatic aberration. We show that both type of components provide a solution to avoid the dead zone and depth ambiguity of SIDFD. Moreover, they imply only a small additional cost and require the tuning of a limited number of parameters. For both approaches, we present a theoretical analysis based on a simple model of the combination of a conventional lens with the add-on. We show experimental results on several types of cameras: an industrial camera, a reflex camera and a smartphone. We evaluate the DFD accuracy with and without the add-on and discuss the potential field of application of the proposed technique. We conclude that both systems improve the DFD estimation process with respect to a conventional camera and that they have different domains of application

A. State of the art

Adding optical components to a conventional camera to improve the performance of the overall system is a widespread approach in computational imaging. In the context of 3D imaging, amplitude and phase masks have been proposed to increase the depth dependence of the PSF [START_REF] Levin | Image and depth from a conventional camera with a coded aperture[END_REF][START_REF] Chakrabarti | Depth and deblurring from a spectrally varying depth of field[END_REF][START_REF] Sellent | Optimized aperture shapes for depth estimation[END_REF] and [START_REF] Dowski | Single-lens single-image incoherent passive-ranging systems[END_REF][START_REF] Greengard | Depth from diffracted rotation[END_REF]. For optimal performance, phase masks have to be put at the principal plane or the Fourier plane of the lens. However, these planes are not precisely known nor easy to access for a commercial lens, lead-ing to non negligible manufacturing costs. Amplitude masks are placed at the camera aperture which is also not always easy to access. Besides, amplitude coded apertures generally do not provide a solution to the depth ambiguity and dead zone of DFD -except for asymmetric apertures [START_REF] Sellent | Optimized aperture shapes for depth estimation[END_REF] which avoid ambiguity.

In previous works [START_REF] Trouvé | Passive depth estimation using chromatic aberration and a depth from defocus approach[END_REF][START_REF] Trouvé | Design of a chromatic 3d camera with an end-to-end performance model approach[END_REF] we have demonstrated the interest of using a lens with a chromatic aberration designed so that three blurs associated to the RGB channels can be measured at each depth. This approach resolves the aforementioned DFD limitations, and allows to limit the amount of defocus blur (at least in one of the three channels) over a wide range of depths. As a result, the accuracy of the DFD was significantly improved with respect to a camera with a conventional lens. However, this approach relies on a specific chromatic lens and thus implies a complex lens design optimization and costs associated to the realisation of tailored optical elements. Besides, this approach can be use only when changing the camera lens is possible.

In the field of particle 3D localisation in microscopy, some authors have proposed to add a cylinder lens in front of the CCD sensor of an inverted microscope [START_REF] Cierpka | On the calibration of astigmatism particle tracking velocimetry for microflows[END_REF]. The cylinder lens acts as a plane-parallel plate in one plane and changes the focal length of the optical system in the perpendicular plane. In doing so, some astigmatism is added to the system and the PSF becomes elliptical, with major axes having a length variable with the depth. By looking at the shape of the elliptical PSF produced on the image plane by a particle one can deduce its 3D position. To the best of our knowledge this approach has never been used for DFD application on natural scenes.

In this paper we propose to add an optical component in front of a conventional camera to improve its 3D measurement ability. The proposed add-ons are simple to realise and can easily be removed from the conventional camera if the user does not need a 3D acquisition. We have looked into two types of add-on: one adds astigmatism thanks to cylindrical lens and the other one adds chromatic aberration with an afocal doublet. Prototypes of the two add-ons have been realized and evaluated through depth estimation experiments using either an industrial camera, a reflex camera or a smartphone. We provide experimental comparison of DFD performance of the system with and without the add-on and discuss the field of applications of each approach.

B. Paper organization

The following sections are dedicated to the cylinder (Section 2) and chromatic (Section 3) components. For each add-on we present its principle, a geometric model that emphasizes the parameter governing the defocus blur and an experimental performance evaluation. Finally, we discuss their respective field of applications. We conclude in Section 4.

CYLINDER ADD-ON

A. Principle

Figure 1(a) illustrates the shape of a cylinder lens. In one plane, referred to as the Oxz plane, this lens acts as a simple planeparallel plate while in the perpendicular plane, referred to as the Oyz plane, it acts as a conventional spherical lens of radius of curvature R. Hence this lens introduces a different position of the focus plane depending on the ray direction, which causes astigmatism. Adding a cylinder lens in front of a conventional camera then leads to an elliptical PSF whose lengths of the main axes depend on depth. Cylinder lenses have been proposed in the literature for particle 3D velocimetry [START_REF] Cierpka | On the calibration of astigmatism particle tracking velocimetry for microflows[END_REF], we study here their use for SIDFD on natural scenes.

B. PSF model

Appendix A presents the model of a lens combining a conventional optical system with an add-on, assuming a Gaussian PSF.

In this section, we use this model to simulate the PSF of an optical system made of a cylinder lens placed in front of a conventional camera.

The cylinder add-on is assumed to be a perfect thin lens with a focal length f a in the Oyz plane and an infinite focal length in the Oxz plane. The value of f a is then:

f a = R n -1 , ( 1
)
where n is the refractive index of the material of the cylinder lens. Here we neglect both the add-on width and the variation of n with respect to wavelength. As the focal length varies in the Oxz and Oyz planes, we propose to model the PSF of the cylinder add-on as an anisotropic Gaussian PSF function :

h(x, y, z) = 1 2πσ x σ y exp - x 2 2σ 2 x - y 2 2σ 2 y , (2)
where σ x and σ y are defined by equation [START_REF] Trouvé | Passive depth estimation using chromatic aberration and a depth from defocus approach[END_REF] in Appendix A with f a = ∞ in the Oxz plane and f a given by equation (1) in the Oyz plane. Figure 2 shows an example of the variation of σ x and σ y with respect to depth for a simulated optical system with f = 35 mm, f # = 4, t px = 9µm and the in-focus plane in the undisturbed direction placed at 1.5 m. The working wavelength is fixed at 530 nm corresponding to the central wavelength of the green channel spectral response of a commercial sensor [START_REF] Jiang | What is the space of spectral sensitivity functions for digital color cameras?[END_REF]. The cylinder add-on has an index of n = 1.5 and is fixed at a distance |e| from the conventional lens arbitrarily put at 15 mm (see Figure 17). Figure 2 shows that the cylinder add-on clearly creates different in-focus plane positions in the Oxz and Oyz planes. The PSF parameters have been traced for three values of the radius or curvature R = 1.5, 2 and 4m: a decrease of R leads to an increase of the distance between the Oxz and Oyz in-focus planes. Hence R appears as the main parameter for tuning the amount of astigmatism of the system.

C. Benefits and limitations

Figure 2 shows that, for a given value of R, there is a single doublet of {σ x , σ y } per depth, thus the ambiguity in depth estimation is resolved. Besides, if no blur variation can be observed in one axis, due to the DoF in this direction, there is always a blur variation that can be observed in the other direction. Hence, the cylinder add-on seems to overcome the main limitations of conventional lens SIDFD. However it is implicitly assumed that both σ x and σ y are observable on the scene. This is not true for mono-directional scenes, ie. edge regions where the intensity is invariant along one direction and follows a 1D profile on the orthogonal direction parametrized by the angle θ in Figure 3 (a). In this case, the observed blur for mono-directional scenes corresponds to the projection of the PSF on the axis parallel to direction θ of the 1D profile. The projected blur size σ proj is then defined as :

σ proj = (σ x cos θ) 2 + (σ y sin θ) 2 .
(3) Note that the proposed add-on usually brings other aberrations in addition to astigmatism. Simulations on an optical design software (assuming a perfect lens for the conventional camera) show that the main unwanted aberration is distortion. This aberration had no noticeable effect in our experiments, but if it had been the case it could have been corrected by distortion calibration and dedicated processing. Other unwanted aberrations can be avoided by a proper optimization of the flat surfaces of the add-on.

In summary, the add-on cylinder offers a way to avoid the classic limitations of SIDFD for scenes with well-distributed gradient orientations. However, it does not bring any gain on one-directional patterns,which appear frequently in man-made environment and particularly in indoor scenes. In such situations, practical use of the add-on cylinder could require additional texture, for instance by using speckle-like structured illumination. Moreover, it should be noted that astigmatism reduces the quality of the image: a deconvolution step can restore it partially, depending on the noise level, at the cost of a significant calculation load however.

D. Experimental validation

D.1. Settings

In the following we experiment the cylinder add-on mounted on an industrial camera, on scenes which are assumed to be sufficiently isotropic to conduct depth estimation. We are interested in depth estimation in the range between 1 m and 1.6m. We use a conventional zoom lens, with continuous variation of the f-number and the focal length. The sensor size is of 1/1.8" and the pixel size t px is of 4.5 µm. The add-on material is in N-BK7 whose refractive index is n = 1.52 at 530 nm. Depth is estimated with and without the add-on, on the green channel extracted from the raw image without interpolation (as a result, the resolution is divided by two and the equivalent pixel size is 9 µm). The focal length and the f-number are approximatively placed at 35 mm and 4 in order to get a defocused PSF of width σ de f between 0 to 3 pixels within the depth range. The focus plane of the conventional lens is fixed at the largest distance of the depth range, ie 1.6 m, to avoid ambiguity when depth estimation is conducted without the add-on. We choose a cylinder lens of radius 2 m, so that both in-focus planes in the Oxz and Oyz planes are contained in the chosen depth range (between 1 m and 1.6 m). Figure 1(b) shows a picture of the resulting optical system.

D.2. Depth estimation algorithm

We estimate the depth using the method described in [START_REF] Trouvé | Single image local blur identification[END_REF] (see Appendix B.A), applied on the green channel extracted from the raw data. For each image patch, a PSF is selected by maximisation of a generalized likelihood among a set of potential PSFs. In practice, this amounts to the optimization of a local cost function over two parameters, the signal to noise ratio (SNR) and the PSF index. Each PSF is related to a depth through a calibration procedure described in the next section.

D.3. PSF calibration

To get a set of potential PSFs as close as possible to the true ones, we calibrate the PSFs from 1 to 1.6 m with a step of 1 cm. Calibration is conducted with and without the add-on using acquisitions of a textured pattern, following the method of [START_REF] Delbracio | The nonparametric sub-pixel local point spread function estimation is a well posed problem[END_REF]. Figure 4 shows examples of calibrated PSFs at different depths with the add-on. As expected, PSFs have an elliptical shape. Note that they are tilted by an angle approximately equal to 45 o , revealing that the major axes of the cylinder lens are tilted with respect to the sensor axes. We experimentally measure the standard deviation σ x,exp and σ y,exp of the blur in both major directions of the cylinder lens. To do so we use a simple knife-edge blur estimation, proposed in [START_REF] Reichenbach | Characterizing digital image acquisition devices[END_REF], on black and white targets positioned at different distances from the camera, having the same orientations as the cylinder major axes. Direct comparison of these experimental values with theoretical σ x and σ y is not straightforward. Indeed we use a zoom lens with a focal length and an aperture that can vary continuously, so the values of f and f # which have been used are only approximation of the true values. Besides, we model the add-on and the conventional lens as perfect thin lenses while in practice they have a physical width. However we show in Figure 5 that one can empirically find a set of parameters f # , e and f , close to the expected values, that gives a good match between theoretical standard deviation and experimental measures of the blur. For this example, those parameters are f = 34.5 mm, f # = 4.5 and |e| = 13 cm. We can conclude that the simple anisotropic Gaussian PSF model deduced from Appendix A and Equation ( 2) provides an efficient model of the hybrid optical system combining a cylinder add-on and a conventional system.

D.4. Experimental depth estimation

Different textured scenes are put at a known distance from the camera. At each depth, a depth map is estimated within a patch of 25 × 25 pixels using the algorithm described in Appendix B. Figure 6 shows the mean depth estimation and standard deviation with respect to the ground truth, inside a centred region of the depth map corresponding to a square of 15 × 15 patches. This experiment is conducted both with and without the cylinder add-on. For both sets of measured values, we have removed 5% of the maximal and minimal values in order to remove outliers in the statistics. Figure 6 shows that in the range 1 m to 1.3 m both configurations (with and without the addon) show unbiased results with a limited standard deviation. More precisely, the mean square error (MSE) is about 2 cm in both cases. After this depth, depth estimation without the addon (middle row of Figure 6) shows large errors with a MSE varying from 8cm to 15cm. Indeed, objects after 1.3 m are sharp because of the camera depth of field, so no blur variation can be measured. In contrast, with the cylinder add-on, the MSE stays below 6 cm in the range 1.4 m to 1.6 m (last row of Figure 6). We conclude that on such textures patterns, the cylinder add-on brings a significant gain in accuracy of the depth estimation.

D.5. Depth maps

Figure 7 shows two depth maps obtained with natural scenes in the range 1 to 1.6 m. Depth estimation is conducted on patches scanning the whole image with a shift of one pixel, thus with maximal overlapping between neighbouring patches. The depth of each object of the scene is correctly estimated if there is a sufficiently contrasted texture. Note however that within regions with mono-directional gradients, such as the region on the top of the cup in (a) there are aberrated results.

CHROMATIC ADD-ON A. Principle

In this Section, we consider another add-on made of a chromatic doublet illustrated in Figure 8. The glasses of each lens are chosen to have an identical refractive index at a nominal wavelength in the visible, but various refractive index at other wavelengths. Thus the add-on acts as a plane-parallel plate for the nominal wavelength, and has a non zero power for the other wavelengths in the visible spectrum, which causes the chromatic aberration. The parameters of the add-on are the two glasses and the radius of curvature R of their common surface. Let n 1 (λ) (resp. n 2 (λ)) be the refractive index of the first (resp. second) glass at the wavelength λ. Using a thin lens approximation, the focal length of the add-on writes:

1 f a (λ) = P a (λ) = n 2 (λ) -n 1 (λ) R . (4)
Figure 8(b) shows an example of power variation of the doublet with respect to the wavelength. The doublet is made of the glasses N-SK16 and N-F2 which have the same refractive index at 577 nm, with R = 20mm. As expected, the power of the doublet is null at the nominal wavelength while it is positive or negative at other wavelengths, which creates chromatic aberration. This figure shows that the optical component acts as a plane-parallel plate for the green channel of the color camera [START_REF] Jiang | What is the space of spectral sensitivity functions for digital color cameras?[END_REF]. Hence the component does not change the main camera parameters for this channel. Note that variation of the refractive index with respect to the wavelength is obtained using Sellmeier equation with parameter from the SHOTT glass catalog.

B. PSF model

The proposed PSFs model of an hybrid optical system made of a chromatic add-on and a conventional lens, is directly based on the model described in Appendix A. Each color channel has isotropic Gaussian PSF parametrized by a standard deviation σ c defined by Equation ( 7) with f a defined for each color channel using Equation ( 4). Note that in order to take into account the intra-channel chromatic aberration, a weighted average of σ is calculated with weights corresponding to the spectrum of each color channel. In the remaining of the paper, the R,G and B spectra are simply modeled using Gaussian functions, with central wavelength and standard deviation respectively equal to (610 nm, 55nm), (530 nm, 35nm) and (470 nm, 30nm) to be globally consistent with the spectra given in [START_REF] Jiang | What is the space of spectral sensitivity functions for digital color cameras?[END_REF]. Figure 9 shows a model of the variation of the PSF standard deviation with respect to depth for the RGB channels and three values of R: 3, 10 and 300 m. The conventional lens main parameters are an aperture of 35 mm with f # = 4 and a pixel size of 12 µm. The in-focus plane is at 3.6 m for the wavelength 532 nm. Figure 9 illustrates that the radius R is the main parameter from which the amount of chromatic aberration can be set. A decrease of R leads to an increase of chromatic aberration. Our aim here is tune the chromatic aberration to have maximally separated color channel DoFs in the range 1 to 5 m. On the other hand, as shown in Figure 9 for R=3mm, too much chromatic aberration leads to higher PSF standard deviations and thus to a decrease of image quality. Given these considerations, for this example, the value R = 10 mm appears as a reasonable choice.

C. Benefits and limitations

As discussed in [START_REF] Trouvé | Passive depth estimation using chromatic aberration and a depth from defocus approach[END_REF] the main advantage of chromatic aberration is that both depth ambiguity and dead zone are avoided, thanks to the separation of the R,G and B in-focus planes. This approach assumes that the scene and illumination spectra are spread enough to have a sufficient SNR in the three R,G,B channels. However this assumption is usually fulfilled for natural scenes and illumination. Now consider the problem of maintaining a good image quality with a chromatic add-on such as the one proposed in this article. First, it yields an increase of both lateral and longitudinal chromatic aberrations. While the longitudinal chromatic aberration can be usefull to estimate depth, the lateral ones induces a scale factor between the R,G and B color channels. Such an effect can be numerically corrected by warping two channels onto the third thanks to a planar homography model. Several efficient techniques are known for such operation. Second, the longitudinal chromatic aberration induces spectrally varying defocus blur. This can be partially corrected using high frequency transfer as proposed in [START_REF] Trouvé | Passive depth estimation using chromatic aberration and a depth from defocus approach[END_REF], a technique which is computationally very efficient. Intra-channel chromatic aberration can also occur and has to be corrected by deconvolution, using the calibrated in-focus PSF of each channel.

Finally, as for the cylinder lens, the add-on can bring aberrations other than chromatic. Simulations conducted on an optical design software (assuming a perfect lens for the conventional camera) shows that the main unwanted aberration is spherical aberration. This aberration had no noticeable effects in our experiments, however it could be reduced if necessary by optimization of the flat surfaces of the add-on.

D. Experimental validations

D.1. Settings

We conduct depth estimation in the range 1-5 m with a camera Nikon D200 having a lens with fixed focal length of 35 mm and a pixel size of 6 µm and f # = 4. Depth is estimated on the color channels extracted from the raw data without interpolation (ie the color channel resolution is divided by two and the pixel size is assumed to be 12 µm). For experiments conducted without the add-on the in-focus plane is put at infinity in order to avoid ambiguity in depth estimation with the conventional lens only. With the add-on the in-focus plane is fixed on the green channel at 3.6 m. The add-on is made of N-SK16 and N-F2 lenses. We built an add-on with R = 10 mm and fixed it in front of a D200 Nikon camera, with a lens of 35 mm, as shown in Figure 10.

D.2. Depth estimation algorithm

SIDFD from RGB images with chromatic aberrations is conducted with the method described in [START_REF] Trouvé | Passive depth estimation using chromatic aberration and a depth from defocus approach[END_REF] and summarized in Appendix B). For each triplet of RGB patches extracted from the raw data, a PSF triplet is selected using a criterion derived within a maximum likelihood framework. Details concerning this algorithm are given in Appendix B. B. Potential PSF triplets were previously calibrated for each depth as described in the next section.

D.3. PSF calibration

With and without the add-on, the PSFs of each color channel are calibrated from 1 to 2.5 m with a step of 5 cm, and from 2.5 m to Fig. 10. Left: chromatic add-on put in the center of a mechanical mount. Right: camera with the chromatic add-on 6 m with a step of 10 cm, using acquisitions of a high frequency pattern [START_REF] Delbracio | The nonparametric sub-pixel local point spread function estimation is a well posed problem[END_REF]. We experimentally measure the standard deviation σ x using a simple knife-edge blur estimation , proposed in [START_REF] Reichenbach | Characterizing digital image acquisition devices[END_REF], on black and white targets put at different distance from the camera. As with the cylinder add-on, direct comparison of these experimental values with theoretical σ is not straightforward. However we show in Figure 12 that one can fit a set of parameters f # , e and f (see Appendix A), close to the expected values, that gives a good match between theoretical standard deviation and experimental measures of the blur. Those parameters are f = 35 mm, f # = 3.9 and |e|=22 mm.

D.4. Experimental depth estimation

Different textured patterns are put at a known distance from the camera. In both cases and at each depth, a depth map is estimated within a patch of 25 × 25 pixels. Figure 13 shows the mean depth estimation and standard deviation with respect to the ground truth, inside a centred region of the depth map corresponding to a square of 15 × 15 patches. We remove 5% of the maximal and minimal values in order to remove outliers in the statistics. With and without the add-on, depth estimation is unbiased in the range 1 m to 5 m. However without the addon the MSE varies between 11 and 85 cm while it varies only between 8 and 40 cm with the add-on. For both scenes the 3D profile within different parts of the scene are correctly restored. In the indoor scene, the estimation fails in textureless and low contrast regions, which is expected from any passive depth estimation technique. In contrast with the cylinder add-on, there is no aberrated values due to monodirectional regions.

Figure 15 shows depth maps obtained with natural indoor and outdoor scenes, with and without the add-on. As already mentioned, whatever the lens, passive depth estimation fails on untextured regions (shown in red). In textured regions, however, the improvement brought by the add-on is very significant. On the two leftmost scenes, the depth maps obtained without addon exhibit higher noise and several of outlier pixels wrongly located at the farthest or closest distance in the used range. The two scenes shown on the right also reveal that the results without add-on suffer from bias. The lower part of the stairs appear too close from the camera and do not exhibit the expected increase of the depth from one stair to the next. On the rightmost scene, the background wall is mislocated at 6m from the camera, while it is actually much closer as shown in the results with an addon. These results confirm that the proposed chromatic add-on allows to reduce noise, outliers and bias on DFD.

E. Applications

Now that we have demonstrated the gain in 3D accuracy brought by a chromatic add-on for SIDFD, we present some applications. First, let us mention that while the depth maps presented in Figure 14 were quite noisy with outliers, they were stemming from a pixelwise decision on the depth value. Using spatial regularization methods one could get much better results, smoother within each object and with better delineation of foreground/background boundaries. As an illustration, we provide in Figure 16 the result of the matting method of [START_REF] Zhuo | On the Recovery of Depth from a Single Defocused Image[END_REF] to the data of Figure 14. Using this type of regularized depth maps opens the way to several applications of the SIDFD using a camera equipped with an add-on. One could think of 3D scanning, man-device interaction, or 3D display.

Another application is refocusing, where a new image is generated from the original image and the depth map, with a different in-focus plane or an enlarged depth of field (the latter case is often reffered to as the "all-focus image"). In Figures 19 we present results of refocusing using a commercial camera and the chromatic add-on. To generate all-focus and refocused images, we first correct for lateral chromatic aberration. To do so we estimate an homography between the R,G and B channels and warp two channels onto the third. Then, denoising is applied using BM3D algorithm [START_REF] Dabov | Image denoising by sparse 3-d transform-domain collaborative filtering[END_REF]. An all-focus image is generated using a high frequency transfer guided by the regularized depth maps of Figure 16. Figure 18 shows one example of such restoration processing. Details on this approach can be found in [START_REF] Trouvé | Passive depth estimation using chromatic aberration and a depth from defocus approach[END_REF]. Note that the wrong depth values have sometimes propagated in textureless regions, however those regions will not create any artefacts in the refocused image, because textureless regions are insensitive to blur. Finally, we produce focused images at different in-focus plane positions using the all-focus image, the regularized depth map and the simple layer model presented in [START_REF] Hasinoff | A layer-based restoration framework for variable-aperture photography[END_REF]. One can then simulate images from a camera with various optical settings in order to increase or decrease the depth of field. For instance, Figure 19 shows the results of refocused images obtained with the Nikon camera with parameters f = 20 mm and f # =3.

Chromatic add-ons can be used with various digital color camera, as long as raw RGB data can be extracted. For instance, Figure 20 shows an add-on designed to be used with a Nexus smartphone.

CONCLUSION

In this paper we have presented two concepts of optical add-ons able to improve depth estimation ability of conventional cameras. The first one adds astigmatism using a cylinder lens and the second one adds chromatic aberration using a doublet of lenses having an identical refractive index only for a nominal wavelength. In contrast with previously proposed unconventional lenses for improved SIDFD, add-ons are cheap and easy to use. We have demonstrated the benefits brought by such add-ons in terms of depth estimation performance compared to conventional cameras. The cylinder add-on requires scenes with isotropic distributions of gradient orientations and leads to difficult image restoration issues. In our opinion, it would be suited to industrial applications where artificial texture can be added by painting or by controlled illumination. The chromatic add-on has less restrictions and can be used on outdoor or indoor situations, with any conventional lens. We have shown that it improves depth estimation and can be applied even for applications dealing with image quality such as image refocusing. Another interesting point about the presented add-ons is that their design depends essentially on a unique parameter, opening ways to efficient optimization process, depending on the application.

Further studies concern the optimization of the add-on with respect to depth estimation accuracy based on a theoretical per-formance model. It could be based on the performance model proposed in [START_REF] Trouvé | Design of a chromatic 3d camera with an end-to-end performance model approach[END_REF] which is a generalization of the model developed for a conventional lens in [START_REF] Trouvé-Peloux | Theoretical performance model for single image depth from defocus[END_REF].

A. PSF MODEL

In this section we present the PSF model of an optical system made of a conventional lens with an add-on, combining diffraction, defocus and pixel integration. One could derive a fine modelisation from the knownledge of principal planes position, as well as input and output pupils positions (that can be measured experimentally) and using for instance Fourier Optics to simulate the PSF. We propose here a simpler model based on thin lens approximation and a Gaussian PSF model.

Both the add-on and the conventional lens are considered as perfect thin lens of focal length respectively f a and f , separated by the distance |e|. Figure 17 shows a scheme of both lenses and presents the main notations. We assume that the global system aperture is still the aperture of the conventional lens. Using the Fig. 17. Schematic view of a hybrid optical system combining a conventional lens and an add-on. thin lens law, the size of the defocus blur de f can be derived

OFig. 1 .

 1 Fig. 1. (a) Cylinder lens. (b) Industrial camera with the cylinder add-on.

Fig. 2 .

 2 Fig. 2. Variation of PSF standard deviation in the Oxz and Oyz planes for different values of the radius of curvature of the cylinder lens. The main parameters of the conventional lens are f = 35 mm, f # =4 and n = 1.5@530nm and the in-focus plane in the Oxz plane is fixed at 1.5 m.

Figure 3 Fig. 3 .

 33 Figure 3(b) shows the variation of σ proj with respect to depth for various values of θ and for the system simulated in Figure 2 with R=1 m. It shows that measuring the projected blur size does not resolves ambiguous depth estimations anymore.

Fig. 4 .

 4 Fig. 4. PSFs at 3 depths: (a) 1m (b) 1.2 m (c) 1.6 m with the cylinder add-on placed in front of a conventional camera.

Fig. 5 .

 5 Fig. 5. Comparison of theoretical and experimental values of σ x and σ y . Theoretical parameters are f = 34.5 mm, f # = 4.5, |e| = 13 cm and R = 2 m and in-focus plane at 1.5m.

Fig. 6 .

 6 Fig. 6. Depth estimation statistical result. First row: pattern used to estimate depth. Second row: without the cylinder addon. Third row: with the cylinder add-on.

Fig. 7 .Fig. 8 .

 78 Fig.7. Experimental scenes and corresponding depth maps using the cylinder add-on. First scene (Top) is made of four objects located at 1.6m (wall), 1.38m (book), 1.2m (pot), 1.03m (doll). Second scene (Bottom) is made from a tilted metallic plate with depth ranging from 1.1m to 1.4m (the hole is at 1.4m) and a picture of leaves on the wall behind at 1.5m. Red is for textureless regions corresponding to α > 0.1 (see Appendix B).

Fig. 9 .

 9 Fig. 9. Examples of variation of RGB PSF standard deviation taking into account intra-channel chromatic aberration, for different values of the radius of curvature of the chromatic add-on respectively R=3, 10 and 300 mm.

Figure 11 Fig. 11 .

 1111 Fig. 11. RGB PSFs at 4 m with the chromatic add-on.

Fig. 12 .

 12 Fig. 12. Theoretical and experimental variations of σ using the chromatic add-on. Theoretical curves are obtained with parameters : f = 35 mm, f # = 3.9, |e| = 22mm and R = 10 mm and a focus at 3.6 m at 530nm.

Fig. 13 .

 13 Fig. 13. Depth estimation experimental statistical results, without (middle row) and with (last row) the chromatic add-on.

Fig. 14 .

 14 Fig. 14. Scenes and depth maps in cm for indoor and outdoor scenes. Red is for textureless regions corresponding to α > 0.1.

  Figures 21 and 22 show experimental image restoration and refocusing obtained with it. Raw data are extracted using a driver developed by the society Codra [22] based on the layer HAL3 of Android 5.0.

Fig. 15 .

 15 Fig. 15. Depth maps in cm for indoor and outdoor scenes. First row color images without the add-on. Second row : depth maps without the add-on, third row depth maps with the add-on. Red is for textureless regions corresponding to α > 0.5.

Fig. 16 .

 16 Fig.[START_REF] Delbracio | The nonparametric sub-pixel local point spread function estimation is a well posed problem[END_REF]. Regularized depth maps using alpha matting[START_REF] Zhuo | On the Recovery of Depth from a Single Defocused Image[END_REF] corresponding to the raw depth map of Figure14.

Fig. 19 .

 19 Fig. 19. Examples of refocused images using the chromatic add-on and the Nikon D200 with a 35 mm lens.

Fig. 20 .

 20 Fig. 20. Chromatic add-on designed for a Nexus 5.

Fig. 21 .

 21 Fig. 21. Top row : Original image taken with the Nexus 5 and the chromatic add-on (left), all-focus image after image restoration (rigth). Bottom row : Zoomed on some image portions (left : raw image, right: corrected image).

Fig. 22 .

 22 Fig. 22. Examples of refocused images with the smartphone Nexus 5 and a chromatic add-on.

from the following equations :

The PSF is modeled as a combined effects of the diffraction, geometric defocus and pixel integration. Assuming that each effect can be modeled as Gaussian PSF function this leads to a combined standard-deviation defined by :

with M the magnification, λ the wavelength, t px the pixel size.

Parameter ρ is a factor empirically fixed at 0.3. To simplify, we fix M = s/z which means that we neglect the (slight) change of magnification due to the add-on.

B. DEPTH FROM DEFOCUS ALGORITHM

DFD consists in estimating depth from one or several defocused images. We note n the number of images used to estimate the depth. In the general case, defocused images are modelled using a local convolution model. Let Y and X be respectively the concatenation of defocused image and scene patch vectors of size respectively nN and nM. One can write :

where N stands for the noise process which is modeled as a zero mean white Gaussian noise (WGN). H z is the block diagonal matrix with each block corresponding to the convolution matrices of the PSFs relative to each image patch. Each PSF can be obtained by simulation or calibration. It is shown in [START_REF] Trouvé | Passive depth estimation using chromatic aberration and a depth from defocus approach[END_REF][START_REF] Trouvé | Single image local blur identification[END_REF] that using a Gaussian prior on the scene parameterized by a precision matrix D, the data marginal likelihood can be derived for a depth and SNR hypothesis. This leads to the following depth estimation criterion:

where P = I -H z H t z H z + αD t D -1 H t z and |A| + refers to the product of the non zero eigenvalues of a symmetric matrix A, with the eigenvalues raised to the power of their multiplicity, m the number of the zero eigenvalues of D. Parameter α > 0 fixes the inverse SNR for the considered patch. k is the index of depth within the finite set of potential depth values {z 1 , ..., z k , ..z K }. Details on the implementation of the algorithm are given in [START_REF] Trouvé | Passive depth estimation using chromatic aberration and a depth from defocus approach[END_REF][START_REF] Trouvé | Single image local blur identification[END_REF]. In the following we briefly describe the contents of H z and D for the two cases considered in the paper.

A. Application to single image DFD

In this case, n = 1 and Y (respectively X) simply collects the N (resp. the M) pixels of the image (resp. scene) patch in the lexicographical order. Matrix H z is directly a convolution matrix which depends on the PSF at depth z. As in [START_REF] Trouvé-Peloux | Theoretical performance model for single image depth from defocus[END_REF][START_REF] Trouvé | Single image local blur identification[END_REF], matrix D is the concatenation of the convolution matrix relative to horizontal and vertical first order derivative kernels [-1 1] and [-1 1] t , which amounts to considering a Gaussian prior on the scene gradients. This matrix as a single zero eigenvalue (m=1) corresponding to homogeneous patch.

B. Application to DFD with chromatic aberration

In this case, n = 3, Y = [y t R y t G y t B ] t , and X = [x t R x t G x t B ] t , where for each channel c, y c (respectively x c ) collects N (resp. M) pixels of the image (resp. scene) patch in the lexicographical order. Given that color channels are highly correlated, vector X is decomposed into luminance and chrominance components. A separable Gaussian prior can then be used for the scene assuming that luminance and chrominance are decorrelated. D is then a block diagonal matrix whose diagonal elements are the concatenation of the convolution matrix relative to horizontal and vertical first order derivative kernels [-1 1] and [-1 1] t , which amounts to considering a Gaussian prior on the luminance and chrominance gradients. Note that the prior matrix D incorporates a parameter that models the ratio between the gradient variances of the luminance and the chrominance components. Matrix H z = H C z T ⊗ I M where I M is the identity matrix of size M × M and T the matrix transforming luminance and chrominances data to RGB data. H C z is the block diagonal matrix with each block corresponding to the convolution matrix of the PSF at depth z relative to each colour channel. Finally, the number of eigenvalues of D is m = 3, for homogeneous patches in the three luminance and chrominances components. Details on this algorithm can be found in [START_REF] Trouvé | Passive depth estimation using chromatic aberration and a depth from defocus approach[END_REF].