Frédéric Magoulès
email: frederic.magoules@hotmail.com

Qinmeng Zou

A Novel Contactless Human Machine Interface based on Machine Learning

Keywords: computer vision, gesture analysis, video analysis, image processing, image segmentation, machine learning, parallel computing

This paper describes a global framework that enables contactless human machine interaction using computer vision and machine learning techniques. The main originality of our framework is that only a very simple image acquisition device, as a computer camera, is sufficient to establish a rich human machine interaction as traditional devices such as mouse or keyboard. This framework is based on well known computer vision techniques and efficient machine learning techniques are used to detect and track user hand gestures so the end user can control his computer using virtual interfaces with very simple gestures.

Introduction

Nowadays, the development of ubiquitous and embedded computing systems in our daily life implies new major challenges in human computer interface design [START_REF] Dix | Human computer interaction[END_REF]. In particular, usual interfaces such as keyboards and mouses are not suited for efficient and convenient communications with such systems. As a consequence, many research efforts are carried to establish a more natural and convenient human computer communication without specific devices. Good reviews on human machine interaction based on computer vision can be found in [START_REF] Cipolla | Computer vision for human machine interaction[END_REF][START_REF] Joseph | A survey of hand posture and gesture recognition techniques and technology[END_REF][START_REF] Moeslund | A survey of advances in vision-based human motion capture and analysis[END_REF]. Futuristic interfaces, as the famous interface presented in the 2002 science fiction film directed by Steven Spielberg entitled Minority Report, have inspired a lot of real advanced interfaces. Most of these futuristic interfaces enable human computer communication at a distance without physical contact but they need specific devices such as hand gloves [START_REF] Sturman | A survey of glove-based input[END_REF] or other tracking devices for human motion capture. Although these devices enable an accurate acquisition of the 3D motion, they are really expensive and cumbersome for real applications.

According to vision based human motion analysis [START_REF] Moeslund | A survey of advances in vision-based human motion capture and analysis[END_REF][START_REF] Poppe | Vision based human motion analysis: an overview[END_REF][START_REF] Wu | Vision based gesture recognition: a review[END_REF] or more particularly on vision based hand gesture interfaces [START_REF] Lenmann | Computer vision based hand gesture interfaces for human computer interaction[END_REF], the different approaches can be classified according to different criteria: (i) the acquisition device (mono [START_REF] Gianni | Interaction visuo-gestuelle avec un mur d'images[END_REF] or multi-camera [START_REF] Utsumi | Hand gesture recognition system using multiple cameras[END_REF] acquisition system), (ii) the gesture representation and tracking approach (articulated kinematic or shape human model [START_REF] Ouhaddi | 3d hand gesture tracking by model registration[END_REF], statistical shape based model [START_REF] Ahmad | Tracking and recognising hand gestures using statistical shape models[END_REF], appearance-based approach [START_REF] Martin | An appearance based approach to gesture-recognition[END_REF], and many other gesture modeling approaches), (iii) the nature of the gesture to recognize (conversational gestures, controlling gestures, manipulative gestures or communicative gestures). Our approaches is voluntarily a simple monocular vision based approach. The design of our framework was made in respect with the following requirements: (i) a single basic camera is used for the computer human interaction (a major choice for the accessibility of the framework in terms of price and infrastructure deployment), (ii) a user friendly reconfigurable interface which is of prime importance for vision based user interface [START_REF] Kjeldsen | Dynamically reconfigurable vision-based user interfaces[END_REF], (iii) a platform and hardware independent framework, (iv) a real time video processing for a rich interaction, i.e., a small latency between the commands given by the end user with hand motions and the execution of the actions on the machine.

The plan of the paper is the following. Section 2 gives a global description of the framework and of its modular architecture. In Section 3, the different modules of the framework are detailed together with some implementation issues. Section 4 presents some experimental results. Finally, Section 5 conclude this paper.

Overview of the Framework

This section presents a global description of our modular framework for the design of contactless human machine interfaces. The different modules correspond to the different tasks needed to build a contactless human machine interface based on computer vision, i.e., (i) a video acquisition module for the capture of hand motions, (ii) a real time video segmentation module for the segmentation of human hand gestures from the video frames, (iii) a hand gesture tracking and recognition module, (iv) an interface module for the communication between the end user and the machine, (v) an engine module to execute the different actions requested by the user via the virtual interface. A global overview of the framework and the orchestration of the different functionalities is described in Figure 1.

The first module is FIZI module. This module transforms each image captured by the camera into a mask of the zones of interest. These connected zones of interest are the skin zones of the end user. Then a selection step enables to select one region corresponding to one hand which is then tracked on all the video sequence. The result of the Tracking module is then sent to Mouse module which maps it to a position on the virtual interactive interface namely the Interface module. At last, Interface module interprets both the position and the area of the tracked zone of interest into actions executed by Engine module. An overview of the complete architecture of the framework is illustrated in Figure 2.

From an implementation point of view, the integration of the framework in the operating system could be achieved through two different approaches. (i) The first approach is the Direct System Integration (DSI) engine which emulates the mouse. This mouse is fully integrated into the surrounding environment. The mouse could for instance control a genuine keyboard and mouse software, such as the visual keyboard. Then Mouse module maps the position into screen coordinates and Interface module does not really exist as a separate entity. (ii) The second approach is the Interface Control (IC) engine which allows to control one particular interface which then could interpret and execute actions.

3 Detailed Description of the Architecture

Functions to Isolate Zones of Interest Module

FIZI module, which stands for Functions to Isolate Zones of Interest, is the module in charge of the video segmentation part. Its main goal is to segment and select the zones of interests in each image of the video sequence. In our case, the zones of interest are the hands of the end user, and to communicate with the machine, the end user has to perform basic hand gestures interpreted by the framework. The FIZI module concerns all the image processing part. The outputs is a binary image which represents the mask of the master hand. FIZI module first runs an initialization step which aims at learning the different parameters that characterize the empty background of the application. Then, three following processing steps are run in parallel: (i) Background removal to select only the relevant information, i.e., the foreground area where the person is, (ii) Grey zone removal by deleting the awkward zones, in order to reduce the luminosity dependance, (iii) Skin color detection and segmentation to select the hands. With these three steps, the framework builds a mask of the different skin zones on each image which could be the hands. A selection enable the selection of the master hand, i.e., the hand used to control-i.e. to give orders-to the machine.

As indicated previously, a first step of FIZI aims at initializing the parameters of the FIZI module by learning the main discriminative features of the background. This learning step consists of a machine learning algorithm similar to the one presented in [START_REF] Zhao | A review on the prediction of building energy consumption[END_REF][START_REF] Zhao | Parallel support vector machines applied to the prediction of multiple buildings energy consumption[END_REF] for energy applications [START_REF] Zhao | Feature selection for support vector regression in the application of building energy prediction[END_REF][START_REF] Zhao | A new parallel implementation of SVM on multi-core systems[END_REF][START_REF] Zhao | New parallel support vector regression for predicting building energy consumption[END_REF][START_REF] Magoulès | Support vector regression for electricity consumption prediction in a building in japan[END_REF] and allows to detect key features of the background [START_REF] Magoulès | Development of an RDP neural network for building energy consumption fault detection diagnosis[END_REF][START_REF] Lai | Vapnik's learning theory applied to energy consumption forecasts in residential buildings[END_REF][START_REF] Zhao | Feature selection for predicting building energy consumption based on statistical learning method[END_REF][START_REF] Zhao | Parallel support vector machines on multi-core and multiprocessor systems[END_REF]. This enable to improve the background removal procedure.

To build the mask efficiently with respect to the need of low computational time, FIZI module runs three different images in parallel. These three images are the same image captured by the camera but in different color spaces: (i) an RGB image is used to remove the background using the learned parameters, (ii) another RBG image is used to remove the grey zones, (iii) an HSV image is used to detect and to segment the skin zones in the image. For each of these images, the processing corresponds to a basic threshold processing.

At last, a merging step is processed to build the output meaningful resulting image. The resulting mask is obtained by a logical AND operator between the three images. Some morphological operations, combining erosion and dilatation operators are applied to remove the small noisy objects and to connect to neighborhood zones. As the output of FIZI module, an image which represent the set of the hand zones of the end user is obtained.

Tracking Module

The objectives of the Tracking module are first, to select the zone of the image corresponding to the guiding hand using image region features. This selection step is done by labeling the region into connected regions of the outputs of FIZI module, i.e., the binary mask of the skin zones, and by the characterization of the different regions. For each connected region, size features such as the area and position features such as the center of gravity and the location in the global image are computed. The different regions are sorted according to these different features. Second the Tracking module tracks the selected region on the video sequence and updates the region features for each frame. A lot of region tracking have been proposed in the literature, and here the tracking is done frame by frame by the described region selection process and a comparison with the previous frame. The output of the tracking module is a image region corresponding to the guiding hand together with its characteristic features.

Mouse Module

Mouse module is the module responsible for building the link between the output of the Tracking module, i.e., a zone of interest in the image corresponding to the guiding hand, with the Interface module. The main goal of the Mouse module is the mapping of this zone of interest into a point or a cursor on the displayed interface. This implies a strong relationship between Interface module and Mouse module. Different mapping approaches are possible to build the between the output of the Tracking module and the Interface module: (i) Absolute mapping: simple ratios are used to build the link. This method is easy and effective when the size of the image frame buffer and the interface are quite close. (ii) Linear relative mapping: the relative displacements are used to move the cursor on the interface. Quite noise sensitive, this method could be tiring for the user. (iii) Non-linear relative mapping: it is the same idea than the previous, but a non linear displacement function is added to ensure small displacements when the user moves its hand slowly and bigger ones when the user moves its hand faster. This method is weakly dependent upon the noise. Figure 3 illustrates these mapping approaches. Absolute mapping is mainly used for Interface Control and for Direct System Integration. For the latter, the non-linear relative mapping is more often considered.

Interface Module

Interface module is the module in charge of the displayed interface. Its functionalities are: (i) to display and to manage the virtual interface from its description in the XML language, (ii) to control, together with the Mouse module, user interactions and associated actions. XML language has been selected to describe these interfaces, mainly because it's user-friendly. XML language allows the users to define the interfaces on their own. The first step is to load and to parse the XML document. The XML file describes the different zones of the interface and how they interact with the system. For instance, a small square could be declared with a text label that emulate the pressing on a key, e.g., 'A' when the user click on the zone. Once the XML file has been loaded, interactions and displays are computed in order to be more efficient while processing one frame. This approach allows to gain some precious milliseconds. In addition, a look-up table is built to quickly determine to which zone belongs the position interacted by the user. Once everything has been processed with the previous initialization computations, the Interface module is ready to be displayed. This entity receives as an input the cursor's position sent by Mouse module, and compute the zone concerned by the current interaction. Then the entity retrieves the actions to be executed and send them to the Engine module.

Engine Module

This last part consists of the engine in charge of the execution process. The processed actions are sent by the Interface module to the Engine module. Engine module is operating system dependent. Its content is quite simple: it sends the system calls corresponding to the received orders. The order are described by three integers: one for the action type and two as parameters.

Implementation

Our framework is developed in C++ with the Open-MP library and the MPI library for the parallel computations. The image processing kernel of the framework is based on OpenCV, since it offers function for easy image manipulation and processing. CBlob library is used for the labeling for the resulting zones after FIZI processing. TinyXml library is used to easily and efficiently parse and load the interface description from an XML file. The code is written in an object oriented approach.

Experimental Results

The first interface available consists of a mouse. All common features of a mouse are present: single left and right clic, double left clic, wheel up and down, moves. The second interface available consists of a keyboard. All the keys from a real keyboard are implemented: letters (A to Z), digits (0 to 9), and special keys (space, backspace, return, etc). Of course, all the keys are not present on the same screen in order to make this interface easier to use with small displacement of the user hand. To select a letter, the right page has to be chosen by the user. Figure 4 illustrates the sequence of gesture for typing the word 'fox'.

Conclusions

In this paper an original approach for contactless human interface is presented. The proposed approach, based on computer vision and machine learning techniques, achieves a virtual mouse and a virtual keyboard using an image acquisition device. Machine learning allows high quality of the captured images, and a parallel implementation ensures fast processing of the captured images. This allows real time interaction of the user with the computer, without physical contact, as required for surgery applications for instance.

H.-X. Zhao the main programmer of the machine learning techniques. The authors acknowledge L. Cabaret and C. Hudelot for the usefull discussions during this long term project. Since 2006, this framework has been named ViKi (Virtual Interactive Keyboard Interface).

Figure 1 :

 1 Figure 1: Global overview of the framework.

Figure 2 :

 2 Figure 2: Architecture overview.

Figure 3 :

 3 Figure 3: Diagram of the mapping approaches. From the left to the right, absolute, relative and relative mapping approach.

Figure 4 :

 4 Figure 4: Sequence of gesture for typing the word 'fox'.

Acknowledgements

The authors acknowledge the numerous students from Ecole Supérieure des Sciences et Technologies de l'Ingénieur de Nancy (France) and from Ecole Centrale Paris (France) who have contributed to this framework since 2003, and in particular N. Vienne, J. Tavernier, the main programmers of the image processing workflow, J. Petin, N. Lambolez, M. Hjalmars, S. Cagnon, C. Mombereau, J. Ott, G. Mathias, S. Massot, D. Miliche, W. Ken, P. Rémi, B. Rochet, J. Holburn, G. Sauwala, A. Brito Alves da Silva, A. Ortega, A. Vinicius Gonzalves Cardoso, F. Mirieu, P. d'Herbemont, E. de Roux, the main programmers of several modules,


Frédéric Magoulès
email: frederic.magoules@hotmail.com

Qinmeng Zou

GPU Accelerated Contactless Human Machine Interface for Driving Car

Keywords: computer vision, gesture analysis, video analysis, image processing, machine learning, parallel computing, graphics processing unit

Introduction

Nowadays, there is not a domain where people don't try to remove wires from computer devices. So, controlling machines at distance, without any physical contact between the human and any object in communication with the machine, is more than what we can hope. Reviews on human machine interaction based on computer vision can be found in [START_REF] Cipolla | Computer vision for human machine interaction[END_REF][START_REF] Joseph | A survey of hand posture and gesture recognition techniques and technology[END_REF][START_REF] Moeslund | A survey of advances in vision-based human motion capture and analysis[END_REF]. Within these systems, contactless human machine have been developed such as hand gloves [START_REF] Sturman | A survey of glove-based input[END_REF] or other tracking devices for human motion capture. Next to wireless mouses and keyboards, technologies such as the virtual keyboard or the WII appeared. Commercialized at the end of the year 2006, the Nintendo' console belongs to the new generation of humanmachine interface. The gamepad is the real innovation of this console. Called Wiimote by Nintendo, the gamepad is indeed like a remote control, connected to the console by the Bluetooth technology. But the Wiimote is more than a simple gamepad. Thanks to an accelerometer, it is able to capture the player's hand's movements: left-right, up-down, forward-backward, rotation, etc. It also incorporates a pointer device, like a computer mouse. And, of course, it can also be used as a normal gamepad when it is horizontally handed. The console offers other high-tech functionalities, while being compatible with former consoles (such as NES, SuperNES, Nintendo64, etc.). It gives, between other things, the possibility of playing directly online. This console is mainly destined for entertainment, and behind the gaming aspect, it brings the possibility of making sport without going out one's room, with the WIISports (Box, Bowling, Golf, Tennis, BaseBall, etc).

With one of our former paper, we have presented previously a framework where the number and the cost of equipment are significantly reduced: only a basic camera is needed, although the same functionalities are available, while opening new perspectives. This framework is an open source framework that enables contactless human machine interaction using computer vision techniques. This framework enables to implement any algorithm requiring a human-machine interaction, without any physical contact. It is developed in C++ and is written in an object oriented approach. Open-MP library and MPI library are used for parallel computations on CPU, and CUDA and OpenCL languages are used for parallel computing on GPU. OpenCV is used for the image processing kernel.

In this paper we present how we have extended this framework for contactless human machine interface for driving car with graphics processing unit accelerated computations. The plan of the paper is the following. Section 2 presents the architecture of the framework. Section 3 discusses some implementations issues of the algorithms. Section 4 illustrate an experiment for the driving. Finally, Section 5 concludes this paper.

Architecture

Our framework works in two steps. The first one consists in the initialization. An XML file is read, containing the configuration to use. Then, the FIZI module learns the background without the user, by using some advanced machine learning techniques. Finally, the interface XML file is loaded. The second step consists in the running one. The camera gets an image, that is treated to deal with the hand's movement. A first treatment is applied to the image so that it can be treated by the program. The brightness is also corrected. Then, the FIZI module can isolate the skin color zones, and more particularly, the hand's one. This hand zone is converted into a pointer by the Mouse module. The Interface module can use this pointer to get its movement, its state (click or not). After this step, it applies the actions associated via the Engine module. These four modules are shortly described in the following paragraphs: FIZI module, Mouse module, Interface module and Engine module.

FIZI module Its aim is to isolate the skin color zones. Therefore, it implements the settings of the luminance and the contour detection parameters, which are easily adjustable to the functioning conditions. After learning the background at the starting of the program, it can isolate a hand.

Mouse module Getting back the binary image from the FIZI module (skin color zones or not), this module tracks the hand corresponding zone.

Interface Module On the two previously mentioned modules, can be added the Interface module of which the role is to load a XML configuration file. This file implements a graphic interface in accordance to the wanted application, linking zones to events. With the Mouse module, it determines the interface zone activated by the pointer. The interface is based on the OpenCV library which makes it independent from the Operating System on which it runs.

Engine module That last module is able to interpret the actions launched by the Interface module, in order to launch system's actions. Opposite to the Interface module, the module is OS dependent.

Workflow A complete description of the workflow is illustrated in Figure 1. Our framework can be adapted for any particular application. The simplest way is to modify only the XML file. But this enables only to adapt the graphical interface and to launch only some basic actions on some events defined with the interface. The other way enables to implement more complex algorithm while modifying directly the main function.

Implementation

Building the Mask of the Hand

To build the mask efficiently with respect to the need of low computational time, FIZI runs three different images in parallel. These three images are the same image captured by the camera but in different color spaces: (i) An RGB image (I 1 ) is used to remove the background using the learned parameters. (ii) Another RBG image (I 2 ) is used to remove the gray zones. (iii) An HSV image(I 3 ) is used to detect and to segment the skin zones in the image. For each of the three images, the processing corresponds to a basic threshold processing, as follow. (i) In image I 1 , pixels which are between the minimum and the maximum values learned in the initialization step are removed and all the others are kept. The resulting image I R1 is a binary image where white pixels represent the foreground and black pixels the background. This corresponds to the values, I R1 (x, y) = 1 if min(I bg min (x, y)) < I 1 (x, y) < max(I bg max (x, y)), I R1 (x, y) = 0 otherwise. (ii) In image I 2 , an interactive threshold operator is applied. For each pixel (x, y) of the image I 2 , if the difference between the minimum min RGB and the maximum max RGB values of the RGB components are lower than a tolerance S defined by the user, the pixel is considered as a gray color pixel and thus is eliminated from the mask. Indeed with high or low luminosity such pixels are often seen as undefined since their colors are arbitrarily set by OpenCV to zero value, i.e., red color, which unfortunately is included in the range of the skin's colors of European people. This corresponds to the values, I R2 (x, y) = 1 if | (min RGB (I 3 (x, y)) -max RGB (I 3 (x, y)) |, I R2 (x, y) = 0 otherwise. (iii) In image I 3 , by using the color value in the HSL color space, the zones with color close to the hand's one can be isolated. These pixels represent a mask of skin zones. For the skin segmentation, two thresholds α1, α2 have to be defined on the Hue space as illustrated in Figure 2. Defaults values have been learn from experiments but the end user can interactively modified them for his convenience and for the customization of the system. These two thresholds are then used to extract the skin zones in images in the following way. For each pixel (x, y) of I 3 , this corresponds to the values:

I R3 (x, y) = 1 if Hue(I 3 (x, y)) ∈ [α 1 , α 2 ]%2π, I R3 (x, y) = 0 otherwise.
At last, a merging step is processed to build the output meaningful result image. The resulting mask is obtained by applying a logical AND operator between the three images. Some morphological operations, combining erosion and dilatation operators are applied to remove the small noisy objects and to connect to neighborhood zones. As the output of FIZI, an image I F IZI represents the set of the hand zones of the end user.

Managing Luminosity

It is the biggest issue of the framework, since thresholds depends on the luminosity of the image. While the application is running, the external conditions are subject to change (dawn or simply clouds modify the quantity of light received by the camera). Since most camera adjust automatically, this leads to brutal changes within exposure time, in luminosity and colors. To correct this we can apply an adaptive threshold or modify the dependent values consequently, as illustrated in Figure 3.

Anyway, the automatic customization can trigger discontinuities in the luminosity leading to mistakes in the hand detection. Luminosity raises issues in the process of the hand isolation by shifting the range of values in the different steps. For instance, in a overexposed environment, the hand is seen by the camera as a white zone but gray zones are removed during the process and thus the hand is not found. A filter has been introduced to modify the Luminosity component in the most obvious cases, as shown in Figure 4 (on the left, the image grabbed from the camera, on the right the image which can then be used as an input for the process).

Acceleration with Graphics Processing Unit

In addition of the existing modules, mathematical operations and linear systems solvers are implemented by Alinea (An Advanced Linear Algebra) library [START_REF] Magoulès | Alinea: An advanced linear algebra library for massively parallel computations on graphics processing units[END_REF]. The Alinea library is implemented in C++, MPI, CUDA and OpenCL. For both central processing unit and graphic processing unit devices, there are different matrix storage formats [START_REF] Ahamed | Fast sparse matrix-vector multiplication on graphics processing unit for finite element analysis[END_REF][START_REF] Ahamed | Iterative methods for sparse linear systems on graphics processing unit[END_REF], and real and complex arithmetics in single-and double-precision. It includes several linear algebra operations and numerous algorithms such as iterative methods [START_REF] Ahamed | Efficient implementation of Jacobi iterative method for large sparse linear systems on graphic processing units[END_REF][START_REF] Ahamed | Conjugate gradient method with graphics processing unit acceleration: CUDA vs OpenCL[END_REF], domain decomposition methods in space [START_REF] Magoulès | Optimized Schwarz method without overlap for the gravitational potential equation on cluster of graphics processing unit[END_REF], together with some energy consumption optimization [START_REF] Magoulès | Green computing on graphics processing units[END_REF].

It is well know that, dynamically reconfigurable vision-based user interfaces [START_REF] Kjeldsen | Dynamically reconfigurable vision-based user interfaces[END_REF] is a key issue to ensure efficient use of any framework. In our framework, dealing with the evolution of the luminosity implies sometimes to re-initiate partly the machine learning techniques which improve the quality of the background removal [START_REF] Lee | Effective Gaussian mixture learning for video background subtraction[END_REF][START_REF] Piccardi | Background subtraction techniques: a review[END_REF]. Autotuning of the GPU memory is thus implemented [START_REF] Magoulès | Auto-tuned Krylov methods on cluster of graphics processing unit[END_REF][START_REF] Ahamed | Energy consumption analysis on graphics processing units[END_REF][START_REF] Magoulès | Power consumption analysis of parallel algorithms on GPUs[END_REF][START_REF] Ahamed | Fast and green computing with graphics processing units for solving sparse linear systems[END_REF] to improve the memory size and computation speed dynamically.

Application for Driving Car

Everybody has at least once played with a car driving game on the computer, probably with a console. The algorithm behind mainly consists of controlling three objects: the left direction, the right direction and the speed of the car. Within our framework, the movements of the hand consists on a rotation around an imaginary wheel. To ease this rotation of the hand, an image of this wheel is added on the captured image by the camera to the screen, so the users knows where are his hand on the virtual wheel. Figure 5 illustrates this interface.

Conclusions

In this paper an original approach for driving car has been presented. The proposed approach achieves a virtual wheel using an image acquisition device, for instance a camera. Interaction of the user with the computer without physical contact by the only mean of a camera has been proposed. Accurate luminosity detection and acceleration of the computations with graphics processing unit, make the proposed framework, a robust, accurate and real time human machine interface. Application for driving car has been illustrated.

Figure 1 :

 1 Figure 1: Architecture of the framework.

Figure 2 :

 2 Figure 2: Threshold values on the Hue space.

Figure 3 :

 3 Figure 3: Hand's color upon the luminosity. On the left, the hand looks violet and on the right two minutes after, the same hand looks blue.

Figure 4 :

 4 Figure 4: Luminosity treatment.

Figure 5 :

 5 Figure 5: Contacless interface for driving car.

Acknowledgements

The authors acknowledge the numerous students from Ecole Supérieure des Sciences et Technologies de l'Ingénieur de Nancy (France) and from Ecole Centrale Paris (France) who have contributed to this framework since 2003, and in particular N. Vienne, J. Tavernier, the main programmers of the image processing workflow, J. Petin, N. Lambolez, M. Hjalmars, S. Cagnon, C. Mombereau, J. Ott, G. Mathias, S. Massot, D. Miliche, W. Ken, P. Rémi, B. Rochet, J. Holburn, G. Sauwala, A. Brito Alves da Silva, A. Ortega, A. Vinicius Gonzalves Cardoso, F. Mirieu, P. d'Herbemont, E. de Roux, the main programmers of several modules, H.-X. Zhao, Sidharth GS, the main programmer of the machine learning techniques, and A.-K. Cheik Ahamed, K. Guilloy, M. Wang, M. Lubin, T. Stremplewski, E. Balagna, I. Petrovic, N. Foresta, N. Fleury, A. Roullier, B. Allardet-Servent, A. Desmaison, F. Mayer, H. Ben Salem, J.-C., T. Saint-Paul, T. Zhu, A. Manai, L. Beaubois, T. Brendlé, O. Darwiche Domingues, O. Soualhine, S. Beux, K. Towa, S. Ehrhardt, S. Olivotto, A. Bodin, M. Lagattu, M. Botarro, E. de Roffignac, C. Haroune, G. Cochet, the main programmers of GPU acceleration techniques. Since 2006, this framework has been named ViKi (Virtual Interactive Keyboard Interface).


Jean Salençon

The influence of confinement on the bearing capacity of strip footings

The bearing capacity for a strip footing acting on a soil foundation with rigid boundaries at a finite distance is investigated within the framework of the Yield design theory both for a purely cohesive soil and for a frictional soil. It is shown that the analysis can be performed simply by referring to already existing results concerning the bearing capacity of a strip footing on a soil layer with limited thickness. Each bearing capacity factor in Terzaghi formula is increased by a factor that increases when the distance of the rigid boundary to the edge of the footing decreases. The bearing capacity is proved to be the more sensitive to this confining effect as the friction angle of the soil increases. From a physical point of view this theoretical analysis would show that, when effective, confinement reduces the depth of the soil layer involved in the collapse mechanism.

soils / bearing capacity / confinement / foundation / Terzaghi / yield design

Effet du confinement sur la capacité portante d'une fondation superficielle Résumé

On étudie la force portante d'une fondation superficielle sur un demi-espace limité horizontalement par des parois rigides. On se place dans le cadre de la théorie du calcul à la rupture. On montre que l'analyse peut être menée simplement à partir des résultats déjà existants au sujet de la portance d'une fondation sur une couche de sol d'épaisseur limitée. Chaque coefficient dans la formule de Terzaghi est multiplié par un facteur qui croit quand la distance de la paroi rigide au bord de la fondation diminue. Cet effet de confinement est d'autant plus sensible que l'angle de frottement du sol est grand. Du point de vue physique cette analyse théorique prouverait que le confinement a pour effet de réduire la profondeur de la couche de sol impliquée dans le mécanisme de ruine. sols / calcul à la rupture / fondation / force portante / Terzaghi

The problem under investigation

The paper is concerned with the theoretical determination of the bearing capacity of a rigid strip footing under axial loading, acting on a homogeneous soil foundation with infinite thickness. The width of the footing is denoted B = 2b. The soil foundation is assumed to be limited in both horizontal directions at a distance L from the edge of the footing by rigid walls. The contact between the footing and the soil on the one side and between the soil and the rigid walls on the other is assumed to be perfectly rough. The goal of the paper is to give a method for assessing the correction factor to be applied to the classical bearing capacity factors to take into account the influence of confinement due to the limited width of the soil foundation as a function of L/b. The theoretical framework is given by the theory of yield design [START_REF] Salençon | An introduction to the yield design theory and its applications to soil mechanics[END_REF] or limit analysis with main purpose of obtaining upper bound estimates for the effect of confinement. The foundation soil is isotropic. It is characterised first by a Tresca strength criterion with cohesion c u , then by a Coulomb strength criterion with cohesion c and friction angle φ.

Strip footing acting on a purely cohesive soil

The problem under consideration

The axial force acting on the footing is denoted F (more precisely, F is the axial force per unit transversal length acting on the footing). The problem is classically treated as 2-dimensional.

The bearing capacity is p ult = F/B. Dimensional analysis proves that for the problem at hand p ult takes the form:

p ult = c u K c L b N c with N c = π + 2 and K c (∞) = 1 ( 1 )
For an unlimited soil foundation (L/b → ∞) the exact value of the bearing capacity is derived from the classical velocity field of the Prandtl solution for the kinematic approach [START_REF] Prandtl | Anwendungsbeispiele zu einem Henckyschen Satz über das plastische Gleichgewicht[END_REF]. For the static approach the stress-fields constructed by Bishop [START_REF] Bishop | On the complete solution to problems of deformation of a plastic rigid material[END_REF] or by Shield [4] or by Sayir and Ziegler [START_REF] Sayir | Zum Prandtlschen Stampelproblem[END_REF] may be used. Fig. 1 sketches the stress field given by Shield and the velocity field. The length of the uplifted wave AD in the velocity field is denoted w and we have w/b = C c = 2.

Static approach

In the case of a soil foundation bounded by rough rigid walls at a distance L from the edges, a static approach is obtained, whatever L/b 0, through considering the restriction of the stress-field in Fig. 1 to the bulk of soil between the walls. The friction condition at the interface between the walls and the soil is fulfilled since the normal stress acting on the rough wall is compressive everywhere: whatever L/b 0 the wall will intersect the three zones I, II, III in the Shield discontinuous stress-field; in zone I, σ yy is constant and compressive: σ yy = -2c u ; zone II is a centred fan where both principal stresses are compressive and therefore σ yy is compressive; in zone III, the principal stresses are σ xx and σ yy which are independent of x and y respectively and such that σ xxσ yy = 2c u ; introducing angle α as indicated in Fig. 1, it comes out explicitly that along the wall in zone III we have -2c u σ yy -2c u (1 + αsin α), which proves that σ yy is compressive. From this static approach it follows that:

∀ L b 0, K c L b 1 ( 2 )

L/b C c

It also appears that the classical virtual velocity field in Fig. 1 is still valid for a kinematic approach as long as L/b C c = 2. From which it follows that:

∀ L b C c = 2, K c L b = 1 ( 3 ) 2.4. 0 L/b < C c
Investigating the range 0 L/b < C c = 2, we refer to the solution given by Johnson and Kudo [START_REF] Johnson | The compression of a rigid perfectly plastic material between rough parallel dies of unequal width[END_REF] and used by Mandel and Salençon [START_REF] Mandel | Force portante d'un sol sur une assise rigide (étude théorique)[END_REF] for the determination of the bearing capacity of a surface footing on a soil layer of limited thickness h, which is presented in Fig. 2. The contact between the soil layer and the rigid bedrock is assumed to be perfectly rough. The slip-line field starts in the same way as in the Prandtl solution and is extended under the footing taking into account the perfectly rough contact conditions under the footing and at the bedrock until it reaches the axis of the footing at an angle of π/4. The geometric ratio w/ h in this slip-line field comes from Prandtl's characteristic fan at point A and is a constant whatever B/ h:

∀ B h , w h = λ c = √ 2 ( 4 )
This slip-line field provides both a velocity field for the kinematic approach and a stress field for the static approach from which the exact value of the bearing capacity of the footing was derived as a function of B/ h with tabulated values of the correction factor F c (B/ h):

p ult = c u F c B h N c = c u F c λ c B w N c = c u F c B √ 2 w N c ( 5
)
In order to obtain an upper bound estimate to the correction factor due to confinement in Eq. ( 1) when 0 L/b < C c = 2, given L/b, we simply consider the Johnson and Kudo slip-line field for which w/b = L/b. The same velocity field as in Fig. 2 provides a kinematic approach for the problem under present investigation (Fig. 3), the whole soil mass beneath SRICD (and sym.) being motionless. (It is worth noting that, in this case, the slip-line field exhibits envelopes for the characteristic lines within the soil mass: since the slip-line field is only used for the derivation of a velocity field, there is no objection to this.) Without any further calculation but referring to the tabulated values of F c (B/ h) an upper bound is obtained for K c (L/b) which is presented in Fig. 3: As expected this upper bound starts from 1 when L/b = C c = 2 and increases monotonically to infinity when L/b → 0. It is worth noting that for 1 L/b 2 the increase of the bearing capacity due to confinement remains below 10%; it exceeds 30% when L/b < 0.56. For comparison sake, in the same Fig. 3, a dotted curve presents the upper bound obtained easily through a simple slip-circle method in the form:

K c L b F c 2λ c L b -1 = F c 2 √ 2 b L (6)
K c L b 0.69(2 + L/b) 2 [(π + 2)L/b] (7)
which proves significantly poorer than Eq. ( 6).

Strip footing acting on a frictional soil

Classical results

Figure 4. Lundgren and Mortensen solution [START_REF] Lundgren | Determination of the bearing capacity of continuous footings on sand[END_REF] and values of N γ (φ) and N c (φ) according to [START_REF] Mandel | Force portante d'un sol sur une assise rigide (étude théorique)[END_REF].

We now address the same problem for an isotropic frictional soil defined through a Coulomb strength criterion with cohesion c and friction angle φ. The bearing capacity of a strip footing on an unlimited soil foundation is usually written following Terzaghi's formula where the theorem of corresponding states is taken into account:

p ult = 1 2 γ BN γ (φ) + cN c (φ) + qN q (φ), N q (φ) = 1 + N c (φ) tan φ (8)
The networks of characteristic lines for the determination of N c (φ) are classical Prandtl's fields for a weightless frictional material. As regards N γ (φ) the solution was given by Lundgren and Mortensen [START_REF] Lundgren | Determination of the bearing capacity of continuous footings on sand[END_REF]: the characteristic network (Fig. 4) proceeds from the classical solution to the problem of the limit equilibrium of a wedge at point A, about which some characteristic features must be recalled. It is self-similar with respect to A, exhibiting no singularity at point A it must be determined through a trial-and-error method starting from the interface under the footing. For the applications to be made it has been necessary to recalculate this solution using the initial values for the trial-and-error method available in [START_REF] Mandel | Force portante d'un sol sur une assise rigide (étude théorique)[END_REF].

As regards N c (φ) the solution can be interpreted as a static approach: the stress field was extended in the whole volume of the foundation soil through Shield's method [START_REF] Salençon | Sur le prolongement statique des champs de Prandtl pour le matériau de Coulomb[END_REF]. It is also a kinematic approach within the theory of yield design since a relevant velocity field can be obtained from the classical equations for the velocity along the characteristic lines in the case of a Coulomb strength criterion. Concerning N γ (φ) the solution was proved to be a kinematic approach (Bonding [START_REF] Bonding | Kinematical admissibility of the pure Nγ rupture figure[END_REF]). The extension of the stress field may still be missing. Fig. 4 gives the values of the bearing capacity factors according to [START_REF] Mandel | Force portante d'un sol sur une assise rigide (étude théorique)[END_REF] for the values of φ that will be considered later on.

When the foundation is a soil layer of limited thickness the same decomposition applies in Eq. ( 9) as in Eq. ( 8) with the introduction of the corresponding bearing capacity factors, functions of B/ h which were determined in the same way as for the purely cohesive soil [START_REF] Mandel | Force portante d'un sol sur une assise rigide (étude théorique)[END_REF]: N γ (B/ h, φ) and N c (B/ h, φ). The characteristic networks are quite similar to Fig. 2 as presented in Fig. 5 for the determination of

N γ (B/ h, φ).        p ult = 1 2 γ BN γ B h , φ + cN c B h , φ + qN q B h , φ N q B h , φ = 1 + N c B h , φ tan φ (9)
The results are available in the form of tabulated correction factors to be applied to N γ (φ) and N c (φ):

N γ B h , φ = F γ B h , φ N γ (φ), N c B h , φ = F c B h , φ N c (φ) (10)

The problem under consideration

The same arguments as above prove that the bearing capacity of a footing on a soil foundation with unlimited thickness and bounded horizontally by rigid walls at a distance L from the edge of the footing is estimated in a conservative manner through a Terzaghi formula similar to Eq. ( 9), where the bearing capacity factors are functions of L/b and φ:

       p ult = 1 2 γ BN γ L b , φ + cN c L b , φ + qN q L b , φ N q L b , φ = 1 + N c L b , φ tan φ (11)
Correction factors K γ (L/b, φ) and K c (L/b, φ) are introduced to account for the influence of confinement:

N γ L b , φ = K γ L b , φ N γ (φ), N c L b , φ = K c L b , φ N c (φ) (12)

Influence of confinement on the N γ term: L/b C γ (φ)

Examining first the N γ term, we consider a strip footing acting on a frictional soil foundation with no cohesion.

The Lundgren and Mortensen solution being characterised with the ratios w/d = λ γ (φ) and w/b = C γ (φ) that are presented in Fig. 6, it follows, assuming the extension of the stress-field has been achieved for this solution, that:

∀ L b C γ (φ), K γ L b , φ = 1 (13)

Influence of confinement on the

N γ term: 0 L/b < C γ (φ)
For the range 0 L/b < C γ (φ), given L/b, we refer to the solution to the problem of the bearing capacity on a soil layer of limited thickness for which w/b = L/b (Fig. 7). The corresponding network of characteristic lines provides a relevant velocity field for the problem under investigation within the framework of the theory of yield design. The corresponding value of p ult is obtained from the first term in Eq. ( 9), by referring to the tabulated values of F γ (B/ h, φ) once the correspondence between L/b and B/ h is known. The solution to the wedge problem being self-similar with respect to point A, the geometric ratio w/ h is a constant whatever L/b < C γ (φ), viz. w/ h = λ γ (φ). Hence the upper bound for the correction factor due to confinement is written:

0 L b < C γ (φ), K γ L b , φ F γ 2λ γ (φ) L b -1 , φ (14)
It is presented in Fig. 7 for some realistic values of φ: 20 • , 30 • , 36 • and 40 • .

Confining effect on the N c term: L/b C c (φ)

As regards the effect of confinement on the N c term in Eq. ( 11), the analysis strictly follows the same path as above, with the simplification that the characteristic line network being derived from Prandtl's solution for the frictional material the geometric ratios w/d = λ c (φ) and w/b = C c (φ) are explicit functions of φ (Fig. 6): Again, it comes out easily that: The assessment of the effect of confinement on the N c term in Eq. ( 11) when L/b < C c (φ) does not need a long comment. Given L/b, we refer to the solution to the bearing capacity problem for the weightless soil layer with limited thickness for which the network of characteristic lines fits exactly within the horizontal bounds of the soil foundation. It provides a relevant velocity field for the problem under consideration and the upper bound for the correction factor K c (L/b, φ) is derived from the tabulated values of F c (B/ h, φ):

w d = λ c (φ) = √ 2 cos φ 2 -sin φ 2 exp π 4 - φ 2 tan φ (15) w b = C c (φ) = 2 (1 + sin φ)/(1 -sin φ) exp π 2 tan φ (16)
∀ L b C c (φ), K c L b , φ = 1 (17)
K c L b , φ F c 2λ c (φ) L b -1 , φ (18)
which is presented in Fig. 8 for the same values of φ: 20 • , 30

Conclusion and comments

The analysis described in the preceding Sections shows that upper bound estimates for the assessment of the confining effect due to the limited horizontal width of the soil foundation can be conveniently made by revisiting classical solutions to the problem of the bearing capacity of a surface footing on a soil layer of limited thickness. From the physical point of view, this means that the presence of the rigid walls, when they are close enough to the footing, reduces the thickness of the zone where the soil is concerned with deformation at collapse.

In the case of the purely cohesive material, the increase of the bearing capacity due to this confining effect remains small for realistic values of L/b.

For the frictional material the conclusions are somewhat different. It comes out clearly from Fig. 6 that both C γ (φ) and C c (φ) are functions which increase quickly with φ. It follows that the bearing capacity is the more sensitive to the influence of the rigid walls which bound the soil foundation as φ increases. Given φ, the correction factors K c (L/b, φ) and K γ (L/b, φ) are functions of L/b which start tangentially to the horizontal axis and increase sharply when L/b decreases to zero. It follows that for realistic values of L/b the increase of the bearing capacity due to the confining effect becomes more and more significant as φ increases. From Fig. 6 we see that in any case C c (φ) is significantly greater than C γ (φ). Comparing Figs. 7 and8 shows that given L/b and φ within the range we have been considering (20 • φ 40 • ), K c (L/b, φ) is, by far, higher than K γ (L/b, φ). One may also notice that in the same time λ c (φ) is but slightly greater than λ γ (φ), which shows that, given L, the thickness of the soil layers involved in the collapse mechanisms for the N c term and for the N γ term respectively would be of the same order of magnitude.

The same method as described in this paper can be applied to the analysis of the effect of confinement on the bearing capacity of circular footings, referring to the solution to the problem of the bearing capacity of circular footings on a soil layer with limited thickness, which was given in [START_REF] Salençon | Capacité portante des fondations superficielles circulaires (Bearing capacity of axially symmetrical shallow foundations)[END_REF].

Figure 1 .

 1 Figure 1. Prandtl solution [2] and extension of Prandtl's stress-field by Shield [4].

Figure 2 .

 2 Figure 2. Strip footing on a purely cohesive soil layer of limited thickness [7].

Figure 3 .

 3 Figure 3. Influence of confinement for a purely cohesive soil.

Figure 5 .

 5 Figure 5. Characteristic network for the calculation of N γ (B/ h, φ) [7].

Figure 6 .

 6 Figure 6. Geometric ratios in the Lundgren and Mortensen solution and in the Prandtl solution.

Figure 7 .

 7 Figure 7. Influence of confinement on the N γ term.

3. 6 .

 6 Confining effect on the N c term: L/b < C c (φ)

Figure 8 .

 8 Figure 8. Influence of confinement on the N c term.