Grégoire Lefebvre

Christophe Garcia

A Probabilistic Self-Organizing Map For Facial Recognition

This article presents a method aiming at quantifying the visual similarity between an image and a class model. This kind of problem is recurrent in many applications such as object recognition, image classification, etc. In this paper, we propose to label a Self-Organizing Map (SOM) to measure image similarity. To manage this goal, we feed local signatures associated to the regions of interest into the neural network. At the end of the learning step, each neural unit is tuned to a particular local signature prototype. During the labeling process, each image signature presented to the network generates an activity vote for its referent neuron. Facial recognition is then performed by a probabilistic decision rule. This scheme offers very promising results for face identification dealing with illumination variation and facial poses and expressions.

Introduction

Over the past two decades, face recognition has been an important research subject in the pattern recognition field that has been extensively investigated. Due to its potential commercial applications, such as surveillance, human-computer interactions, vision systems and video indexing, identifying human faces remains a challenging problem. The main difficulties are due to illumination constraints, facial expressions and orientations. Whereas holistic matching methods use the whole face region and face feature-based methods consider local regions as the eyes, nose and mouth, we investigate the "bag of features" representation [START_REF] Csurka | Visual Categorization with Bags of Keypoints[END_REF] which models an object by a set of local signatures. Based on interest point detection, we assume that the relevant salient biometric information is sufficiently redundant whatever view is considered. For each salient point, we focus on its near influence area to describe the signal singularity. The edge descriptor should then compute a stable signature, regarding geometric transformation. This large amount of training information is then organized thanks to a Self Organizing Map [START_REF] Kohonen | Self-Organizing Maps[END_REF]. A decision rule based on conditional probability is then defined, using a learning by example strategy from facial feature stimulation on SOM neurons. This paper is organized as follows. In section 2, we first present our face recognition scheme based on SOM learning from local descriptions. Section 3 is presents some experimental results that illustrate the performances of the proposed method. And finally, conclusions are drawn.

Supervised face classification scheme

System architecture

As underlined in [START_REF] Duda | Pattern Classification[END_REF], a classification scheme is generally composed of three main steps: pre-processing, feature extraction and classification. In our approach, the pre-processing step consists in detecting some salient points in the image to be compared, reducing thus the zones of interest to a limit number of regions considered as perceptually relevant. From each detected salient point, a salient patch is extracted and a local feature vector is calculated. Each local feature vector is then fed into a SOM network resulting in a neural activity map composed of all winning cells, as shown in our previous study [START_REF] Lefebvre | Supervised image classification by SOM activity map comparison[END_REF]. Finally, to complete the feature extraction step of our classification scheme, the obtained activity map is used in order to label each SOM unit by measuring the frequency of SOM prototype appearance, individual by individual. Assessing the similarity between a test image and an individual model is then reduced to compute a new decision rule from a maximum a posteriori probability (MAP) using previous labeled neuron stimulations. The different computational steps used in this method are detailed in the next sections.

Regions of interest description

According to the active vision mechanisms [START_REF] Hoffman | The Role of Visual Attention in Saccadic Eye Movements[END_REF], the goal of salient point detectors is to find perceptually relevant image locations. Many detectors have been proposed in the literature [START_REF] Harris | A Combined Corner and Edge Detector[END_REF][START_REF] Bres | Detection of Interest Points for Image Indexation[END_REF]. The salient locations selected by human visual system contain generally high contrast, lines and edges. Following this observation, we focus our interest on the salient point detector [START_REF] Laurent | In Depth Analysis and Evaluation of Saliency-based Color Image Indexing Methods using Wavelet Salient Features[END_REF] that uses a Haar wavelet analysis to find pixels on sharp region boundaries. The facial features are then described in some regions of interest around each salient point (cf. Figure 1). Many local descriptors have been proposed [START_REF] Schmid | Local Grayvalue Invariants for Image Retrieval[END_REF][START_REF] Lowe | Distinctive Image Features from Scale-Invariant Keypoints[END_REF][START_REF] Bay | Surf: Speeded up robust features[END_REF], but we chose the Regularity Foveal Descriptor [START_REF] Laurent | Description of local singularities for image registration[END_REF] using foveal wavelet to describe the 2D signal singularity. Using again wavelets is justified by the consideration of the human visual system for which multi-resolution, orientation and frequency analysis are of prime importance.

SOM learning

For face recognition, we learn a global SOM from all face signatures, without considering each identity (cf. Figure 2). The Kohonen model [START_REF] Kohonen | Self-Organizing Maps[END_REF] is based on the construction of a neuron layer in which neural units are arranged in a lattice L. Usually, the lattice is two dimensional (rectangular or hexagonal). The neural layer is innervated by d input fibers, called axons, which carry the input signals and excite or inhibit the cells via synaptic connections. The Kohonen network aims at preserving the topology of the input space and at tuning each cell to a particular set of stimuli. To reach these goals, the excitation of neurons has to be restricted to a spatially localized region in L and the location of this region has to be determined by those neurons that respond most intensively to a given stimulus. Moreover, L acts as a topographic feature map if the location of the most strongly excited neurons is correlated in a regular and continuous fashion with a restricted number of signal features of interest [START_REF] Ritter | Neural Computation and self-Organizing Maps -an introduction[END_REF]. Neighboring locations in L correspond thus to stimuli with similar features. To satisfy these properties, a neighboring function between cells must be added in the network model. For this purpose, each cell i ∈ L is connected to a set N L (i) of neighboring cells, defining thus a topological ordering. The goal of the Kohonen learning algorithm is then to adapt the shape of L to the distribution of the input vectors. The 2D lattice shape changes to capture the input information and the topology existing in the input space. Those two properties can be considered as a competitive learning and a topological ordering. Let M be the input space and X = x(t), t ∈ {1, 2, . . . , T } be a set of observable samples with x(t) ∈ M ⊂ ℜ d , t being the time index. Supposing M = m i (t), i ∈ {1, 2, . . . , N } and t ∈ {1, 2, . . . , T } is a set of reference vectors with m i (t) ∈ ℜ d , randomly initialized. The best matching unit (BMU) m c (t) is then defined by the index c :

c = arg min i ||x(t) -m i (t)||, ∀i = 1, . . . , N × N (1)
A kernel-based rule is used to reflect the topological ordering observed in the human visual cortex. The updating scheme aims at performing a stronger weight adaptation at the BMU location than in its neighborhood. This kernel-based rule is defined by :

m i (t + 1) = m i (t) + λ(t)φ (i) c (t) [x(t) -m i (t)] (2)
where λ(t) designates the learning rate, i.e. a monotonically decreasing sequence of scalar values with 0 < λ(t) < 1 and φ (i) c (t) designates the neighborhood function that governs the strength of weight adaptation as well as the number of reference vectors to be updated. Classically, a Gaussian function is used, leading to :

φ (i) c (t) = exp - ||r c -r i || 2 2δ(t) 2 (3)
Here, the Euclidian norm is chosen and r i is the 2D location for the i th neuron in the network. δ(t) specifies the neighborhood width decreasing during the time.

SOM labeling

In the following, let us use the following notations to build a SOM labeling process : N × N the number of SOM neurons; K the number of individuals; T the number of local signatures for an image; P the number As shown on Figure 3, we label the previous learning SOM with the activation of each local signatures from the learning image database. That is to say that we focus on each learning signature projection on SOM lattice and we store each best matching unit stimulation, class by class. Consequently, at the end of the labeling process, each SOM unit is labeled with the number of time that it has been activated as a winner neuron for each individual. This label value is defined by the equation 4:

L k (n) = A k (n) K k=1 A k (n) (4) A k (n) = P l=1 card{Z l (n)} (5) Z l (n) = {x(t) ∈ I l , ||x(t)-m n || < ||x(t)-m j ||, j = n} (6)

Decision rule

The decision rule consists of maximizing the a posteriori probability P (C i /I t ) to deduce the identity of the individual I t (cf. Figure 4 and Equation 7). Following the Bayes's theorem and the equiprobability of each individual belonging, we obtain the equation 8. Using the local signature independence for a test image, the probability P (I t /C i ) can be deduce from its local signature projections on SOM lattice with the associated learning label element L k (n) as in Equation 9.

I t ∈ C i ⇐⇒ i = arg max k P (C k /I t ) (7)
P (C k /I t ) = P (I t /C k ) K q=1 P (I t /C q ) , P (C k ) = 1 K (8)
P (I t /C k ) = N ×N n=1 card{Z t (n)} × L k (n) (9)
Z t (n) = {x(t) ∈ I t , ||x(t)-m n || < ||x(t)-m j ||, j = n} (10)

Experimental Results

For all the experiments, we configure our SOM with the following rules to reach good learning results from 500 RFD individual signatures in term of accurate input data representation [START_REF] Kohonen | Self-Organizing Maps[END_REF] : T = 500 × N ; λ(t) = T /(T + 99t); δ(t) decreases linearly from √ 2N/2 to 1/2. In these experiments, we focus on two face databases (cf. Figure 5) : The first database named ORL1 is collected by AT&T and Cambrigde University Laboratories. 40 distinct subjects are available with 10 image samples. For some subjects, the images were taken at different times, varying the lighting, facial expressions (open / closed eyes, smiling / not smiling) and facial details (glasses / no glasses). The second database, YALE 2 , contains 165 views of 15 persons. The 11 face images per person present illumination variations and facial expressions as happy, sad, sleepy or surprised. In our experiments, all faces are extracted using the face detector proposed in [START_REF] Garcia | Convolutional face finder: A neural architecture for fast and robust face detection[END_REF] and resized to 200 × 200 pixels. In order to evaluate the system performances, we use a 3-fold cross validation method in the following experiments.

Figure 6 presents our method result with different SOM size and Table 1 proposes a comparison with wellknown face recognition algorithms published by Yang et al. [START_REF] Yang | Kernel eigenfaces vs. kernel fisherfaces: Face recognition using kernel methods[END_REF] who use a leaving-one-out cross validation. Then, it is very interesting to observe that our proposal challenges the statistical methods with respectively 98.67% and 94.44% of good identification on the ORL and YALE databases. The respective standard deviation are 0.12 and 4.15. These performances appear

Conclusion

In this paper, we propose an original face recognition system using directly local signature information. Based on the two main properties of SOM, which are dimension reduction and topology preservation, this architecture features all facial identities by neural activity counts. In order to quantify the visual similarity between a test image and the global neural model, we build a probabilistic decision rule. This solution implemented for facial recognition gives us very promising results. However, a growing and pruning strategy or a hierarchical SOM could be useful to determine automatically the SOM size from learning data.

Figure 1 .

 1 Figure 1. Facial feature description.

Figure 2 .

 2 Figure 2. SOM learning process.

Figure 3 .

 3 Figure 3. SOM labeling.

Figure 4 .

 4 Figure 4. Decision rule.

Figure 5 .Figure 6 .

 56 Figure 5. ORL and YALE image samples

Table 1 .

 1 Classification rate comparison.when the SOM size achieves 1225 units. With this configuration, the learning SOM process clusters more precisely the different face signatures and the individual frequency labeling proposes a better classification result. From a size of 225 neurons, we perform more than 80% of good recognitions but when this number is upper than 1225, we are confronted to the classical overlearning issue. Increasing the number of neurons means growing the SOM learning time but the decision result is still immediate, independently to the number of persons and the learning examples.

	Approach	ORL	YALE
	ICA 3	93.75%	71.5%
	Eigenfaces	97.5%	71.5%
	Kernel eigenfaces 98%	75.8%
	Fisherfaces	98.5%	91.5%
	Kernel fisherfaces 98.75%	93.9%
	Our method	98.67%(±0.12) 94.44%(±4.15)

3

Independent Component Analysis

http://www.cl.cam.ac.uk/research/dtg/attarchive/facedatabase.html

http://cvc.yale.edu/projects/yalefaces/yalefaces.html


Bastien Rizzon
email: bastien.rizzon@univ-smb.fr

Vincent Clivillé
email: vincent.cliville@univ-smb.fr

Sylvie Galichet
email: sylvie.galichet@univ-smb.fr

Pascal Ochalek Elodie

Ratajczak Adixen
email: elodie.ratajczak@adixen.fr

Pfeiffer Vaccum

Decision Problem of Instrumentation in a Company involved in ISO 50001

In the context of performance management and the involvement of companies in sustainable development, energy management is a new challenge. Managing energy is a complex problem as it can potentially impact the whole company and if mishandled could be harmful for its added value. In order to assist companies in this challenge, the ISO 50001 proposes a standard for the creation of Energy Management Systems (EnMS). Such systems bring the recurring problem of decision about the choice of the actions necessary into their deployments. Notably, they rely heavily on the data related to the consumptions of energy and its usage in a given company. Hence the problem of decision, at an audit stage, can concern the choice of a sound instrumentation. This paper deals with the interest of the company adixen Vacuum Products in a standard for decision aiding in this situation. The idea is to build the model of preference of the Decision Maker and to establish a generic procedure about decision supported by a MCDA tool. Our proposition is to use the ACUTA method in order to elicit this model of preference.

I. INTRODUCTION

At the start of the 21st century, energy savings and the limitation of greenhouse gas emissions have become major challenges for many organizations including companies. Thus the International Standard Organization (ISO) estimates the potential gain to be about 30% on the energy consumption. In this context the ISO edited a specific norm for the companies involved in the control of their consumptions of energy through a system of management: the ISO 50001, published in 2011 [START_REF]Win the energy challenge with iso 50001[END_REF] .

The expectations and the major role of instrumentation for such a system of management are now presented.

A. The ISO 50001 concerning the Instrumentation

The ISO 50001, Energy Management Systems Requirements with guidance for use, gives organizations the requirements for Energy Management Systems (EnMS) as a standard. It provides a framework of requirements enabling organizations to:

• Develop a policy for more efficient use of energy

• Fix targets and objectives to meet the policy • Use data to better understand and make decisions concerning energy use and consumption

• Measure the results

• Review the effectiveness of the policy

• Continually improve energy management.

According to [START_REF] Vikhorev | An advanced energy management framework to promote energy awareness[END_REF] "ISO 50001 includes requirements for a program of energy data collection such as the installation of Automatic Monitoring and Targeting (AMT) systems, and this can be used to increase energy efficiency at different levels of a factory". This can be seen as the application for an EnMS of the well known principle of "What you measure is what you get" [START_REF] Kaplan | The balanced scorecard: Measures that drives performance[END_REF] [START_REF] Lorino | Méthodes et pratiques de la performance: le guide du pilotage[END_REF].

Usually, a company has knowledge about one point of measurement for a given energy: a single global point available directly with a counter or indirectly through the related bills. Intuition or practical observations point to the lack of information in such a single point of measurement. There is also the possibility of taking the reading directly on an equipment. This is generally considered as a timeconsuming activity with many possibilities for errors. However the exhaustive instrumentation of equipments and machines in order to measure their consumptions of energy is expensive, time consuming, detrimental for the company's activities of production or R&D,. . . For example such an instrumentation in order to get the electrical consumption in real time of a single equipment can cost from 600e to 1500e. So, a company which cannot afford to "measure everything" has to plan for the instrumentation of its activity in order to select the most relevant measurement points.

Before presenting the instrumentation decision problem, let us recall some key points about the general framework of improvement in companies, the so called Quality Management System (QMS) which is very similar to the EnMS. Indeed according to [START_REF]Win the energy challenge with iso 50001[END_REF], the ISO 50001 is built with the same common elements as the other ISO standards, including the ISO 9001 which deals with QMS and is well known by many companies.

B. Quality Management System

According to the ISO 9001, the QMS is based on the PDCA cycle often illustrated by the Deming's wheel [START_REF]Qualité et systèmes de management ISO 9000[END_REF]. More precisely, improvement must be made Step by Step, must be related to the company's processes according to its Process Mapping. Every action must be controlled before launching the next step. In this way the PDCA four stages are itemized as follows:

• Plan: the establishment of the objectives and the development of the action plan,

• Do: the deployment of the action plan,

• Check: the measurement of the results with regard to the planned objectives,

• Act: the corrections and improvements of the plan and its deployment (before the next cycle).

We call

Step one cycle of the Deming's wheel. These steps are usually marked with the adoption of standards in order to capitalize on the experience gained in the process of Continuous Improvement. In order to control the successive steps of the improvement, Process Managers need decision aiding.

In this study, decision aiding concerns the instrumentation of equipments in order to provide energy consumptions related to process and activities to their managers. The proposed idea is to provide a decision aiding framework easily usable for all the steps of the improvement. Let us detail this problem of decision in the next subsection.

C. A Decision Problem for Instrumentation

The decision process has to establish a balance with:

• the connection between the electrical network and the Process Management in the company,

• the degree of accuracy in the energy consumption mapping,

• the costs in time, resources,. . . allocated to the instrumentation.

The company is interested in standardizing this process in order to replicate it for other perimeters of its activity. During this process of adopting the standards of the ISO 50001 that deal with energy management (which impacts all the company's activities) the rigorous and systematized process is also defined. In order to deal with this problem, this paper is based on the case presently studied in the company adixen Vacuum Products (aVP). The idea is to build with MCDA 1 approaches of the MAUT 2 family [START_REF] Jacquet-Lagreze | Preference disaggregation: 20 years of experience[END_REF] the model of preference of the DM for the problem of decision concerning instrumentation. More precisely a method based on preference disaggregation for their holistic characteristics is adopted. Amongst these methods, ACUTA [START_REF] Bous | Acuta: A novel method for eliciting additive value functions on the basis of holistic preference statements[END_REF] is appreciated by the company for its interactivity.

This paper is organized around two main sections. Section II presents aVP and its situation with regard to energy management. It also shows how the deployed instrumentation for a part of the electrical network of the company has been matured since it had an influence in the decision making 1 MultiCriteria Decision Analysis 2 Multiple Attribute Utility Theory process. Section III presents the deployment a posteriori of a MCDA method, namely ACUTA, in order to build the model of preference of the DM for this particular question of instrumentation. Then thus model could be used for the next step(s) of improvement.

II. PRELIMINARY ANALYSIS OF ENERGY CONSUMPTIONS

A. Presentation of the Company aVP

Since 2011, aVP has been a French subsidiary of the Pfeiffer Vacuum Technology AG Group. The group's primary activity is to offer solutions based on vacuum technology for a broad range of applications, including the semi-conductor, analytic instrument, coating and R&D markets, as well as a variety of general manufacturing sectors. These applications are represented in aVP's sectors of activity, notably:

• productions,

• supports to production,

• R&D,

• services.

These sectors of activity rely on a large know-how in the field of mechanics and electronics and the infrastructure related built from the 1950's to the present days. Certified ISO 9001 and 14001, aVP is now aiming for the certification ISO 50001 in alignment with its management organization. In order to support the management of its sectors of activity, the company has purchased and installed a software infrastructure that allows to build custom, Web-enabled applications for accessing, automating and controlling "smart" devices in real-time over the Internet. Now that the activity of the company is described from a management point of view, the next subsection will focus on the energy consumed, starting from a financial point of view.

B. Existing Data about Energy Consumptions

The company is supplied with two types of energy:

• Electricity

• Gas

It is important to note that there is no interaction between these two types of energy. Indeed for aVP gas is only involved for comfort heating and electricity is not involved in heating in the company. So the two types can be discriminated in the EnMS without the risk of missing a potential interaction. A fast analytical work based on the bills related to these energies has shown that about 90% of the total of the energy costs are related to electricity. Thanks to this information the DM has been able to argue for an approach that should focus on electricity initially.

However the single global value of electric consumption emitted every month on the bill is not sufficient in terms of specification (considering the different sectors of activity, buildings,... in the company). So it is relevant to "go beyond" the global point of view. The global electrical station distributes electricity to 8 substations. Each substation is already instrumented with electrical counters that allow to record the consumptions related to each substation. A fast analytical work based on these data has shown the distribution of electricity amongst the 8 substations (see figure 1). According to this result the substation P is the major point of consumption.

The next stage now is to check what is beyond the substation P and to figure if actions can be decided from this point of view. To achieve this we proceeded to specify the equipments related to the substation and the existing readings.

It has appeared quickly that it was impossible to follow the electricity consumption of an equipment and its related sector of activity from the readings of the substation. Substation P distributes electricity for equipments in areas that were progressively built during the last decades. So this electrical network distributes electricity for machining, assembly lines, services, design and research,. . . So it is very difficult to match the energy consumptions with the Process Mapping which is mandatory for the EnMS because the future energy consumption gains will be managed by the Process Managers. Furthermore there is no instrumentation existing in the equipments related to substation P.

For the EnMS the DM wants to develop something close to the Management by Process. However the diverse sectors of activity and so their management are mixed in the substation P counter. In order to decide on the adequate instrumentation the first thing was to map the network of equipments related to substation P with the input by expertise of the equipments known as consumptive and their related sector of activity.

C. Instrumentation for the Electric Substation P

Figure 2 is the illustration of the Electrical Network of Substation P represented as an organizational tree:

• The root of the tree (level 1) is the electrical cabinet which distributes electricity to the many elements of substation P.

• The leaves of the tree (level 2 to 5) are the equipments or machines supplied by the substation P.

• The electrical consumption read on a Parent Node is the direct sum (plus non-insignificant loss) of the consumptions of its Children Node (with the exception of one generator illustrated by its own symbol in the tree).

The nodes are considered as points of interest for the EnMS which can all be instrumented. The nodes are categorized depending of the sector of activity concerned with the equipment. When all the children of a node belong to the same sectors of activity, the parent node is categorized the same way. When the children of a node are of different sectors of activity, the parent node is considered as a Mix node.

The nodes noted as important consumers were identified by expertise. The other nodes are either non important consumers or the expert could not assess them.

Supported by this new information the DM decided for an instrumentation (see section III-A6). He/she conducted meetings and brainstorming in order to take into consideration the many constraints in the project of instrumentation. The problem was handled without the support of a MCDA tool.

The idea presented in the next section is to build the model of preference that would have been elicited if a MCDA approach had been deployed. By doing so, the next step(s) in the process of Continuous Improvement (the instrumentation of substation H5 for instance) would thus in this way benefit from this model in its decision process a priori.

III. ELICITATION OF THE MODEL OF PREFERENCE

In this section, the ACUTA method will be deployed on the practical case of the instrumentation of the most consumptive electric substation in the company.

The method is run a posteriori of the decision for this instrumentation. However the process here is to rebuild the considerations of the DM when he/she decided for a specific instrumentation into a Multicriteria Decision Problem. By deploying ACUTA on this problem the DM would get information about the rigor behind the decision and to improve the process of such a decision for the next action plans. In this paper the rigor to be expressed is the balance operated between the different criteria considered in his/her process of thought for deciding which alternative to implement for the company's EnMS. This balance will be expressed in the model of preference.

The different alternatives and criteria in this decision problem will now be detailed with their motivations.

A. Description of the Alternatives 1) Initial Instrumentation (a01): This is the state of the instrumentation before the deployment of the new instruments. This is considered as an alternative for the DM who could chose to Do nothing if the other alternatives were not worth the effort. Furthermore it is representative of a "global" measurement framework.

This instrumentation consists of a single point for measurement at the root of the electrical network (see figure 2). 2) Exhaustive Instrumentation (a02): The idea of this alternative is to provide information related to electrical needs in the most accurate way by monitoring every final element of electrical consumption in the network.

By adding these values, according to the structure of the network, every node can be controlled. This instrumentation consists of the deployment of points of measurement for every leaf in the network (see figure 2).

3) Sub-global Instrumentation (a03):

The idea of this alternative is to provide information related to electrical needs for the level in the network just below the global point of view. This alternative is considered by the DM who wants to assess if going "one step further" of the global point of view would be satisfactory. This instrumentation consists of the deployment of points of measurement for every node in level 2 of the network (see figure 2).

4) Sectors of Activity Instrumentation (a04):

The idea of this alternative is to provide information related to electrical needs that can be linked to the different sectors of activity in the company. This alternative is considered by the DM because it would be easier to make the link between a dysfunction in electrical consumption and the related sector of activity. of measurement, according to the structure of the network, in a way that allows to discriminate measurements as belonging to a given sector of activity. It is illustrated by figure 3 with grayed nodes as instrumented nodes.

This instrumentation consists of the deployment of points

In practice: if a parent node can be classified in another category than MIX it is to be instrumented; if not check for the children nodes and so on.

5) Production Centered Instrumentation (a05):

The idea of this alternative is to provide information related to electrical needs for the added value activity of the company. This alternative is considered by the DM because it highlights a critical aspect of the strategy of the company: Balancing its energetic needs with the quality of its lucrative activity.

This instrumentation consists of the deployment of point of measurements on the node related to the Production activity of the company, trying as much as possible to be accurate and is illustrated by figure 4.

In practice: every leaf node belonging to the Production or Production Related categories are to be instrumented (represented by a grayed node).

6) Handpicked Instrumentation (a06):

The idea of this alternative is to provide information related to electrical needs for a set of nodes that the DM chose one by one. This alternative is considered by the DM who wants to consider a non-systematic approach for the instrumentation while focusing on the most consumptive equipments (identified This instrumentation consists of the deployment of points of measurements on the node selected for reasons specific to the company's needs. It can be the result of a process of selection during a brainstorm and is illustrated by figure 5 with grayed nodes as instrumented nodes. This is the list of the alternatives in the problem, now let us see about the criteria.

B. Description of the Criteria

1) Compatibility with the Governing Organization of the Company (g1): This criterion allows for the assessment of the concerned alternative according to the ability to give information about electrical consumptions directly to the manager concerned.

The criterion is defined from α 1 = 0 to β 1 = 10 as a scale. In order to build this scale, the DM must verify that the score increases progressively according to his/her satisfaction. The higher the value is, the closer to the governing organization of the company the instrumentation is. This is a Gain criterion as the DM is more satisfied when the instrumentation allows the breakdown of electrical consumptions into something similar to the governing organization of the company.

2) Total Distance between the Points of Measurements (g2):

This criterion allows the assessment of the concerned alternative according to a technical constraint. In this case study, the technical constraint is about the cabling of the different points of measurements for the alternatives concerned.

The criterion is defined from α 2 = 0m to β 2 = 200m. Every 100m a repeater has to be installed in the cabling. Furthermore the disturbance for the activity of the company is higher when the action of instrumentation concerns a bigger area. The DM does not want more than 1 repeater to be installed, so the upper limit for this criterion is 200m. This is a Cost criterion as the DM is more satisfied when the installations concerned are closer to each other.

3) Durability of the Instrumentation (g3): This criterion allows the assessment of the alternative concerned according to the longevity of the physical equipments to be monitored. The machine can be replaced or displaced which is detrimental for the DM because it would be necessary to reconfigure the cabling after such an important modification. This criterion is defined from α 3 = 0 months to β 3 = 24 months. 0 means that the equipment or the machine is to move "today" while 24 means that the equipment or the machine is not to move before 24 months at least . This is the upper limit for this criterion as such action plans are seldom defined beyond a 2 year horizon. This is a Gain criterion as the more durable the equipment or machine will be, the more satisfied the DM is.

4) Expectation in a Gain for the Energy Management System (g4): This criterion allows the assessment of the alternative concerned according to the expected outcome in order for the EnMS to be built. Data about this criterion are uncertain because the evaluation of the future gains is a costing and difficult task. So in fine this criterion remains "qualitative".

The criterion is defined with a reasonable linguistic scale of 3 terms: low, medium, high. For the sake of simplicity in this case study these linguistic values are turned into the numeric values 1, 2, 3, i.e α 4 = 1 and β 4 = 3. This is of course a Gain criterion.

5) Total Cost (g5):

This criterion allows the assessment of the concerned alternative according to the total cost of the instrumentation which is self explicit.

The criterion is defined according to a fixed price of 700 e per point of measurement. If the DM is constrained by a budget limit some alternatives could be immediately discarded through this criterion. This is of course a Cost criterion ranging from α 5 = 0 e to β 5 = 47600 e Now that we have described the alternatives and the criteria, the next subsection will deal with the different inputs required for running ACUTA with the Diviz software [START_REF]Diviz-decision deck[END_REF].

C. Inputs

Figure 6 is an illustration of the input for using the ACUTA method through Diviz of the Decision Deck [START_REF]Diviz-decision deck[END_REF].

The inputs named alternatives and criteria are simply the identification of the alternatives (here a 01 to a 06 ) and the criteria (here g 1 to g 5 ).

The input named preference Direction is about the nature of each criterion as a Gain or Cost criterion. All these inputs were already figured in the III-A and III-B. Let us see about the others.

1) performanceTable: This is the Decision Matrix (see table I) where the values describe each alternative from the point of view of the 5 criteria of the problem. These values are obtained from estimations, measurements, linguistic expressions,. . . 2) alternativesRank: This is the order of reference established by the DM. The DM expressed his/her preference directly over the actions, such as for example "I prefer each other alternative over the exhaustive instrumentation; I prefer the level 2 instrumentation over initial instrumentation" and so on. In the end, the preferences of the DM were expressed with the following reference order: a 06 a 05 a 04 a 03 a 01 a 02 3) criteriaSegments: This input concerns the number of segments for each value function. In this case study, the number has been set to 2 for all criteria.

So the model of preference will take the form of five 2piece linear value functions as g i (a k ) → u i (g i (a k )) with their breakpoints at γ i such as

γ i = αi+βi 2 .
The ACUTA method can now be run in order to compute a model of preference. It is important to note that the model of preference is not unique, but ACUTA elicits a single model of preference by trying to respect at best all the constraints of the problem at once (see [START_REF] Bous | Acuta: A novel method for eliciting additive value functions on the basis of holistic preference statements[END_REF] for more details). There are other approaches that deal with the elicitation of a single "representative" model of preference with preference disaggregation methods (see [START_REF] Siskos | Uta methods[END_REF]). A considered benefit for the choice of ACUTA is the location of the solution, "as far as possible" from the boundary of the constraints polyhedron.

D. Output: valueFunctions

This is the model of preference of the DM for this problem of instrumentation. This model is synthesized in table II and illustrated in figure 7.

The corresponding utility given to the alternatives can be seen in table III.

The DM can now figure the relative importance he/she had in mind for the different criteria according to the Decision Matrix and his/her preferences. From this model of preference the utility of an alternative given its evaluation according to the five considered criteria can be calculated.

E. Discussing the Model of Preference

Hence it is possible to discuss with the DM about the validity of the model of preference. The DM could either keep the model of preference as it is, in order to see if it can be considered as robust before testing it in the next instrumentation. Or the DM could disagree with the model of preference according to his/her judgment and so it would provoke questions about the Decision Matrix, the enrichment of the information given on the DM's preferences, changing the number of pieces in the value functions, . . . Assuming that this model is reliable, it could now be used to assess different or future alternatives in comparison to each other. Let us consider the alternative Energy Consumptive Equipments Instrumentation a 07 . This instrumentation consists of the instrumentation of the equipments judged by expertise as consumptive. The evaluation of the alternative for each criterion is as follows: g 1 (a 07 ) = 7; g 2 (a 07 ) = 100; g 3 (a 07 ) = 20; g 4 (a 07 ) = 2; g 5 (a 07 ) = 4200

With the model of preference the utility for this alternative is 0, 61, and so it is preferred only to a 02 and a 01 . But the model of preference itself is not the only gain in this approach.

F. Acquired Experience

The whole approach deployed for this problem of instrumentation can now be standardized for the next step in Continuous Improvement. This standardization at a strategic level is illustrated by the figure 8. with the DM about the model of preference, the company has acquired some knowledge it was not directly seeking in the instrumentation problem. For example the Electrical Network scheme from a management point of view is now a document of support for mapping the company's activity more accurately. In order for the criterion g 4 to be correctly assessed we carried out information processing on existing data related to energy consumptions of aVP. This has allowed to identify a dysfunctional equipment and to understand how it should be dealt with. This experience has helped to refine the framework that the DM wants for the EnMS.

IV. DISCUSSION

We proposed in this paper a practical approach for aiding a DM to obtain a more rigorous point of view on his/her own decision. The case studied here dealt with the required instrumentation to build an EnMS as a sound management system according to the related official standards. In order to do so, a multicriteria model of preference has been elicited by interacting with the DM and the company's existing data. This model is still to be refined and improved. For instance the preponderant criterion g 4 could be turned into a "quantitative" criterion when the gain for the EnMS can be measured and analyzed after the complete implementation of the instrumentation. Despite the potential weakness of the model of preference in its present state, the fact of building it from the making of the problem of decision is considered to be rich in information and experience. The approach has been adopted as a standard by aVP and after discussion is to be generalized for the other substations or adapted for other problems of decision concerning management systems.

However this standard was developed in a favorable situation with no interaction between the types of energy used by the company. Furthermore the huge difference between the importance of the types of energy in this case study allowed for a simple discrimination in the realization of the standard proposed here. The approach should be further developed in order to extend it for other companies or organizations as suggested by the ADEME -French Environment and Energy Management Agency. In this context it would require to study more general standards involving the different types of energy consumed by organizations.

Fig. 1 .

 1 Fig. 1. Electric Substations Relative Consumptions

Fig. 2 .Fig. 3 .

 23 Fig. 2. Electrical Network Illustration

Fig. 4 .

 4 Fig. 4. Illustration for Instrumentation a05

Fig. 5 .

 5 Fig. 5. Illustration for Instrumentation a06

Fig. 6 .

 6 Fig. 6. The Toolbox ACUTA on Diviz

Fig. 7 .

 7 Fig. 7. Plot of the Model of Preference

Fig. 8 .

 8 Fig. 8. Adopted Standardized Strategic Approach for Energy Monitoring in aVP

TABLE I .

 I THE DECISION MATRIX FOR THIS PROBLEM OF INSTRUMENTATION

		g1	g2	g3	g4	g5
	a01	0	0	24	Low	0
	a02	10	200	6	Medium	47600
	a03	7	10	20	Medium	7700
	a04	9	120	12	High	16100
	a05	8	80	6	High	19600
	a06	6	5	20	Medium	4200

TABLE II .

 II THE MODEL OF PREFERENCE COMPUTED WITH ACUTA

	gi	αi	ui(gi(αi))	γi	ui(gi(γi))	βi	ui(gi(βi))
	g1	0	0	5	0,10	10	0,13
	g2	0	0,25	100	0,13	200	0
	g3	0	0	12	0,03	24	0,08
	g4	1	0	2	0,10	3	0,32
	g5	0	0,23	23800	0,07	47600	0

TABLE III .

 III UTILITY OF THE ALTERNATIVES COMPUTED WITH ACUTA-DIVIZ

	Alternatives	Utilities
	a01	0,55
	a02	0,25
	a03	0,69
	a04	0,7
	a05	0,71
	a06	0,71

ACKNOWLEDGMENT

Presented PhD work has been developed thanks to the financial support of ADEME (Agence de l'Environnement et de la Maitrise de l'Energie/ French Environment and Energy Management Agency) and APS (Assemblée des Pays de Savoie) and the participation of adixen Vacuum Products.


Julien Ros

Christophe Laurent

Grégoire Lefebvre

A Cascade of Unsupervised and Supervised Neural Networks for Natural Image Classification

This paper presents an architecture well suited for natural image classification or visual object recognition applications. The image content is described by a distribution of local prototype features obtained by projecting local signatures on a self-organizing map. The local signatures describe singularities around interest points detected by a wavelet-based salient points detector. Finally, images are classified by using a multilayer perceptron receiving local prototypes distribution as input. This architecture obtains good results both in terms of global classification rates and computing times on different well known datasets.

Introduction

With the dramatic increase of available digital contents, advanced content management solutions become essential. If we focus on the particular situation of digital images (Infotrends1 expects that the number of images captured on camera phones will reach 227 billion by 2009), efficient images management solutions such that supervised image classification have to be found. The goal of a supervised image classification system is to group images into semantic categories giving thus the opportunity of fast and accurate image search. To achieve this goal, these applications should be able to group a wide variety of unlabelled images by using both the information provided by unlabelled query image as well as the learning databases containing different kind of images labelled by human observers. In practice, a supervised image classification solution requires three main steps [START_REF] Duda | Pattern Classification[END_REF]: pre-processing, feature extraction and classification. Based on this architecture, many image classification systems have been proposed, each one distinguished from others by the method used to compute the image signature and/or the decision method used in the classification step. Regarding the signature computation, the most efficient methods are probably the local aproaches firstly introduced in [START_REF] Schmid | Local grayvalue invariants for image retrieval[END_REF]. In this case, local signatures are computed around some interest points and their values are chosen in a dictionnary obtained from the training database. Local signatures are used to represent the image by a distribution of local image features easily classifiable as in [START_REF] Csurka | Visual categorization with bags of keypoints[END_REF][START_REF] Jurie | Creating efficient codebooks for visual recognition[END_REF] or are directly used to learn a model used for the next recognition step [START_REF] Weber | Unsupervised learning of models for recognition[END_REF][START_REF] Fei-Fei | A hierarchical bayesian model for learning natural scene categories[END_REF][START_REF] Marée | Random subwindows for robust image classification[END_REF]. In the state of the art, the dictionnary is classically computed thanks to a Kmeans algorithm [START_REF] Weber | Unsupervised learning of models for recognition[END_REF] or by a bottom-up clustering procedure [START_REF] Agarwal | Learning to detect objects in images via a sparse, part-based representation[END_REF]. We propose here to use a self organizing map [START_REF] Kohonen | Self-Organizing Maps[END_REF] to generate the visual dictionnary. Furthermore, in our approach, a Multilayer Perceptron classifier is built with the training dataset and is used for the last classification step. The paper is organized as follows. Section 2 describes the method which was introduced earlier in [START_REF] Laurent | In depth analysis and evaluation of saliency-based color image indexing methods using wavelet salient features[END_REF] to detect interest points and extract local image features. Section 3 presents the self-organizing map algorithm, the construction of the vocabulary of local descriptors and the construction of the image feature vector. The design of the Multilayer Perceptron classifier and the decision rule are explained in detail in section 4. Experiments are presented in section 5 and finally, section 6 concludes the paper.

Local Features Extraction

The goal of feature extraction is to reduce the amount of data contained in an image by extracting relevant and discriminating features. In local approaches, this extraction phase results in feature vectors computed around interest points and an image I j is thus represented by a set of local signatures S(I j ) = {s 1j , . . . , s nj }. It is important to mention here that local approaches result in a lack of ordering between signatures.

Interest Points Detection

The goal of interest point detectors is to find image locations that are perceptually relevant for the next recognition step. Many detectors have been proposed in the literature, each one focusing on a particular local property of the image content such as contrast [START_REF] Bres | Detection of interest points for image indexation[END_REF], corners [START_REF] Harris | A combined corner and edge detector[END_REF][START_REF] Mikolajczyk | Scale and affine invariant interest point detectors[END_REF], edges [START_REF] Laurent | In depth analysis and evaluation of saliency-based color image indexing methods using wavelet salient features[END_REF][START_REF] Loupias | Wavelet-based salient points for image retrieval[END_REF], etc. The salient points detector presented in [START_REF] Laurent | In depth analysis and evaluation of saliency-based color image indexing methods using wavelet salient features[END_REF] uses a wavelet analysis in order to find relevant pixels located on sharp region boundaries. The use of wavelet analysis is motivated by observing that multi-resolution, orientation and frequency analysis are of prime importance for the human visual system during the recognition step. This detector has proven its efficiency in many vision applications [START_REF] Laurent | In depth analysis and evaluation of saliency-based color image indexing methods using wavelet salient features[END_REF] and thus will be used in the present work.

Description of Local Singularities

Most local descriptors describe the local neighborhood of salient points by characterizing edges in this area. Edge information thus appears fundamental in the process of local neighborhood description. To describe edges, gradient orientation and magnitude are generally used. Nevertheless, from a mathematical point of view, an edge or more generally a singularity can also be efficiently characterized by considering its Hölder exponents. We propose to use this mathematical notion to design our local descriptor.

Definition 1. f : [a, b] → R is Hölder α ≥ 0 at x 0 ∈ R if ∃K > 0, δ > 0 and a polynom P of degree m = ⌊α⌋: ∀x, x 0 -δ ≤ x ≤ x 0 + δ, |f (x) -P (x -x 0 )| ≤ K|x -x 0 | α .
Definition 2. The Hölder exponent h f (x 0 ) of f at x 0 is the superior bound value of all α. h f (x 0 ) = sup{α, f is Hölder α at x 0 }.

The local regularity of a function at a point x 0 is thus measured by the value h f (x 0 ). It is worth noting that the smaller h f (x 0 ), the more singular is the signal at the point considered. For example, the Hölder exponent of a Dirac impulse is -1 and 0 for a step function. For an image, the Hölder exponent is measured in the direction of the minimal regularity of the singularity (in the gradient direction). The different singularities met in an image are shown on figure 1. To

Original Image -0.2 ≤ h ≤ 0.2 -1.2 ≤ h ≤ -0.8
m(x, y) 2 = (I j (x + 1, y) -I j (x -1, y)) 2 + (I j (x, y + 1) -I j (x, y -1)) 2 (1)
θ(x, y) = tan -1 I j (x, y + 1) -I j (x, y -1)

I j (x + 1, y) -I j (x -1, y) . (2)
Then, for each singularity, the Hölder exponent h is estimated with foveal wavelets as presented in [START_REF] Mallat | Foveal Approximations for Singularities[END_REF]. Orientations and Hölder exponents maps are then conjointly used to construct different 3D histograms. To build such histograms, each ROI is first partitionned into 4 × 4 blocks and each histogram is computed in a particular block before being normalized by the block size (See figure 2). This last step of the signature design is realized in the same spirit as the construction of the SIFT descriptor presented in [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF]. Finally, the signature is obtained by concatenating the different 3D histograms and thus has a size of n × r × o where

Image Representation

To use a classical machine learning methods for the last classification step, images must be represented by a vector of equal size. When local image descriptors have been extracted, a powerful and recent method is to represent the image by an histogram of local descriptors, this is the "bag of keypoints" representation introduced in [START_REF] Csurka | Visual categorization with bags of keypoints[END_REF]. Nevertheless, it supposes that local descriptors are quantized into a visual dictionnary of fixed size. We propose to build such a dictionnary by using a self-organizing-map.

Self Organizing Map Learning

The self-organizing map (SOM) is an unsupervised classification algorithm based on competitive learning [START_REF] Kohonen | Self-Organizing Maps[END_REF]. It is a variant of the k-means algorithm that has the advantage of preserving the topology of input datas X = {x(t), t = 1, 2, . . .} with x(t) ∈ D ⊂ R n and providing thus a better description of them.

The SOM aims at projecting the input data space D into a lower dimensionnal space (1D, 2D,. . . ) defined by a regular discrete lattice L composed of N nodes. Therefore, it is a vector quantization algorithm which preserves the topology of the input space because each node c of the lattice is a neuron with a codebook vector w c ∈ R n such that if c 1 and c 2 are close then w c1 and w c2 are close in R n . For this purpose, the SOM is trained thanks to a competitive learning algorithm Fig. 3. General Principle of a Self Organizing Map which supposes first that the SOM has been correctly initialized. For example, w i (0) could have been initialized randomly ∀i = 1, 2, . . . , N . At epoch t of the learning step, x(t) is compared simultaneously to all w i (t) by using a distance measure d(x(t), w i (t)) on the input space D and the best candidate vector w c (t) associated to the node c (the best matching unit or BMU) is chosen such that:

w c (t) = arg min i d(x(t), w i (t))i = 1, 2, . . . , N. (3)
The learning scheme uses then a kernel based rule to update the weights:

w i (t + 1) = w i (t) + α(t)h ci (t)[x(t) -w i (t)] (4)
where 0 < α(t) < 1 is the monotically decreasing learning rate. Furthermore, h ci denotes a neighborhood function that governs the strength of weight adaptation as well as the number of reference vectors to be updated (generally, a gaussian function is used). It is worth noting that a good choice for the number of iterations during the learning is 500 times the number of cells in the SOM.

Bag of Local Descriptors Representation

As previously emphasized, at this stage of the algorithm, an image I j is described by a set of local signatures S(I j ) = {s 1j , . . . , s nj } representing 3D histograms around interest points presented in section 2.2. Thus, this kind of representation could not be directly interpreted by a classifier because of the lack of ordering between signatures. Moreover, the number of signatures could be different for two images (due to different number of interest points detected). Thus, to build and use an image classifier, the image I j should be represented by a feature vector H(I j ) = [h 1j , . . . , h N j ]. For this purpose, an indexing step should be used to transform the set of local signatures into a precise and compact representation of the image content. This is a classical problem met in text categorization where a document composed of a set of words has to be characterized by a vector describing its content. For this purpose, a text is often represented by a vector of term weights, where the terms are chosen in the codebook (a set of meaningful words for the understanding of texts); this is the well known "Bag of Words" representation. This approach has influenced the work presented in [START_REF] Csurka | Visual categorization with bags of keypoints[END_REF] and denoted "Bag of Keypoints" which proposed to adapt text categorization methods to the computer vision problems. Similarly, we propose to represent the image content by the probabilistic distribution H over local images features. This distribution is in fact the activation histogram of the SOM previously learned. For this purpose, each local signature of the image activates a particular cell (The BMU) and participates to an update of the histogram H(I j ). The bins h lj are defined as follow:

h lj = card{s k ∈ I j , ||s k -w l || < ||s k -w i ||∀l = j, k ∈ {1, . . . , n}}. (5)
At this stage of the algorithm, each image is represented by a unique feature vector denoted H. The natural image classification problem is thus reduced to a multi-class supervised classification problem. For this purpose, we have tested a multilayer perceptron (MLP), a Radial Basis Function network classifier (RBF) and a Support Vector Machine classifier (SVM). Experimentally MLP exhibited better results than the RBF and equivalent results than the SVM. Thus we restrict our discussion on this classifier. MLPs can be used for classification problems and are multi layers feedforward neural networks fully connected. Thanks to their fundamental property of parcimonious approximation, they are well suited to modelize any continuous function g : R N → R p , where N is the dimension of the input space and p is the number of classes. However, it supposes that sufficient neurons are chosen during the definition of the network architecture. A three layer perceptron architecture with N intputs, n h hidden neurons and p output neurons is presented on figure 4.

Fig. 4. General Principle of a Multilayer Perceptron

For an input data X = [x 1 , . . . , x N ] ∈ R N , the output of the k th output neuron of this MLP is given by the discriminant function:

g k (X, W ) = ϕ   n h j=1 w kj ϕ N i=1 w ji x i + w j0 + w k0   ∀k ∈ {1, . . . , p} (6)
where W = {w ij } is the set of weights of the neural network considered and ϕ is the activation of the neurons of both the hidden and ouput layer. This one should be non-linear allowing the MLP networks to model nonlinear mappings well and is the standart sigmoidal function in the following. MLPs are trained with the backpropagation algorithm which adapts the weights of the network in W to their optimal values for the given pairs (X l , t(X l ))∀l ∈ {1, . . . , N L } in the training dataset. If the target vector t(X l ) gives the correct class of X l and is such that t k (X l ) = 1 if X l ∈ C k and 0 otherwise then the trained network approximates the correct a posteriori probabilities concerning the classification problem: g k (X, W ) ≈ P (C k |X). An input data X can thus be easily classified by regarding the maximal value of the output neurons.

Experimental Results

In this section, the system scheme is tested on different well known datasets. The experiments particularly emphasize on the influence of the vocabulary size(i.e. the size of the SOM) on the classification results. These results are analyzed by evaluating global classification rates. Furthermore, the analysis of ROC curves by computing the area under curve (AUC) for the best parameters will be presented providing a direct comparison to other algorithms. It is worth noting that the MLPs used in the experiments have a number of nodes in the hidden layer which is the mean of the number of nodes in the input and ouput layer n h = N +p 2 . It to achieve good classification results in reasonable computing times.

Presentation of the Datasets

The first dataset is extracted from the SIMPLICITY database2 . It contains 500 images of size 384 × 256 and is divided into a learning dataset of 250 images and a test dataset of 250 images. There are five clusters: beaches, buildings, buses, elephants and flowers as shown on figure 5.

Influence of the Self Organizing Map Size

The influence of the vocabulary size on the classification results is conjointly tested with the number of interest points extracted from the image. The SOM is small dictionnary of 250 prototypes. Nevertheless, the results between 15 × 15 and 10 × 10 are not so different compared from those obtained with a 5 × 5 lattice. Moreover, a study has shown that the quantization error does not decay very fast if the SOM is larger. It is thus not necessary to improve the dictionnary size.

For the two dataset, 3000 interest points are a good compromise in term of computing times and global classification rates. In this case, the method obtains a global classification rate of 86.4% for the SIMPLICITY dataset (86.4% for the SVM and 82% for the RBF) and 82.9% for the PASCAL database (81.7% for the SVM and 75% for the RBF).

ROC Curve Analysis and Confusion Matrix

On figure 8, AUC and confusion matrix are displayed for the parameters exhibited in the previous section. For the PASCAL dataset, the results are good and comparable to those obtained during the PASCAL 2005 recognition challenge.

Computing Times

The system must be efficient both in classification results and in computing times in order to be attractive for a human. Whereas features extraction, SOM and MLP learning are realized offline and could thus be long, the classification

Conclusion

This paper presents a neural network architecture for natural image classification using local images features. It has been shown that a self organizing map could learn a small visual dictionnary subsequently used to represent the image content by a distribution over the prototypes. Moreover, a classification step based on a multilayer perceptron has shown to be efficient. The approach exhibits high classification rates and small computing times. Its implementation in a professional application is thus possible. The perspectives are to use the Growing Hierarchical Self Organizing Map to generate the codebook [START_REF] Rauber | The growing hierarchical self-organizing maps: Exploratory analysis of high-dimensional data[END_REF] and to represent the image as a "bags of graphs" generated by grouping interest points which could be learned thanks to a SOM for structured datas [START_REF] Hagenbuchner | A self-organizing map for adaptive processing of structured data[END_REF].

Fig. 1 .

 1 Fig. 1. Different Type of Singularities

Fig. 2 .

 2 Fig. 2. Principle of the Singularity Descriptor

Fig. 5 .

 5 Fig. 5. Images from the SIMPLICITY Database

Fig. 6 .Fig. 7 .

 67 Fig. 6. Images from the PASCAL Database

  AUC and Confusion Matrix for the SIMPLICITY and Pascal Dataset of a query image must be fast. The computing times obtained on a Pentium IV with 3Ghz are shown on figure9. Moreover the features extraction for the entire training set takes 290s for the SIMPLICITY database and 1301s for the PASCAL database. The training steps (SOM and MLP learning) are not too long and so totally realistic in an offline mode. It is worth noting that SOM learning does not depend on the database size and the interest point number but only on the dimension of the SOM as emphasized in section 3.1 because the number of iterations is 500 times the number of cells in the SOM. The classification of a new instance is very fast and could thus be realized online in a professional application.

	Class	AUC	Beach Buildings Buses Elephants Flowers classified as
	Beaches 0.9125	37	6		2	3	2	Beach
	Buildings 0.9330	8	37		0	4	1	Buildings
	Buses 0.9959	2	1		47	0	0	Buses
	Elephants 0.9844	4	1		0	45	0	Elephants
	Flowers 0.9971	0	0		0	0	50	Flowers
	Class	AUC		Bicycles Cars Motorbikes People classified as
	Bicycles 0.926		71	23	14	6	Bicycles
	Cars	0.9622		7	252	5	11	Cars
	Motorbikes 0.9793		7	3	202	4	Motorbikes
	People 0.8862		13	18	7	46	People
	Fig. 8. Dataset	SIMPLICITY			PASCAL
	Size of the SOM	5 × 5 10 × 10 15 × 15 5 × 5 10 × 10 15 × 15
		SOM learning	4s	46s	185s	5s	46s	195s
		MLP learning	12s	127s	665s	30s	413s	1706s
	MLP classification 0.00012s 0.00064s 0.00244s 0.00012s 0.00073s 0.00232s
			Fig. 9. Computing Times

http://www.infotrends-rgi.com/home/Press/itPress/2005/1.11.05.html

http://wang.ist.psu.edu/ jwang/test1.tar

http://www.pascal-network.org/challenges/VOC/


Grégoire Lefebvre
email: gregoire.lefebvre@francetelecom.com

Huicheng Zheng
email: zhengh@tcd.ie

Christophe Laurent
email: christophe2.laurent@francetelecom.com

Objectionable Image Detection by ASSOM Competition ⋆

This article presents a method aiming at filtering objectionable image contents. This kind of problem is very similar to object recognition and image classification. In this paper, we propose to use Adaptive-Subspace Self-Organizing Maps (ASSOM) to generate invariant pornographic features. To reach this goal, we construct local signatures associated to salient patches according to adult and benign databases. Then, we feed these vectors into each specialized ASSOM neural network. At the end of the learning step, each neural unit is tuned to a particular local signature prototype. Thus, each input image generates two neural maps that can be represented by two activation vectors. A supervised learning is finally done by a Normalized Radial Basis Function (NRBF) network to decide the image category. This scheme offers very promising results for image classification with a percentage of 87.8% of correct classification rates.

Introduction

In many computer vision applications such as multimedia data mining, pattern recognition, image retrieval, etc., evaluating image content is fundamental. Recognizing harmful images is very challenging in content-based filtering systems.

In the state-of-the-art, different approaches are proposed, focusing on skin color detection. Forsyth et al. use geometric constraints for detecting naked people [START_REF] Forsyth | Identifying nude pictures[END_REF] by reconstructing the human anatomic structure from skin areas. The WIPE system [START_REF] Wang | Classifying objectionable websites based on image content[END_REF] combines Daubechies wavelets, moment analysis and histogram indexing to provide semantically meaningful feature vector matching. Based on the discrete probability distributions obtained from skin and non-skin histograms, Jones and Rehg [START_REF] Jones | Statistical color models with application to skin detection[END_REF] filter images according to their skin pixel statistics. Other studies [START_REF] Bosson | Non-retrieval: Blocking pornographic images[END_REF][START_REF] Zheng | Blocking adult images based on statistical skin detection[END_REF] propose a pornographic image detection system based on skin detection and Multi-Layer Perceptron (MLP) classification.

To avoid inherent skin detection problems, as illumination variations, background interferences, multiple figures, etc., we consider objectionable image filtering as image classification problem in two categories: adult and benign. Image classification consists in partitioning the input image space into a number of regions separated by decision surfaces and labeled by image classes.

For a given image I, the ultimate goal is to search for a function f (•) → J I , I → J, where I is the image space and J the image label space. However, due to the extremely high dimension of an ordinary image, a direct search of the optimum function f (•) in the original image space I would generally not be possible.

From this observation, we try to describe the image I in a more compact way to reduce the dimension of data and obtain the most discriminant informations. In this purpose, we are interested in generating invariant-feature descriptors by using ASSOM neural networks [START_REF] Kohonen | Self-Organizing Maps[END_REF].

Adaptive-Subspace Self-Organizing Map (ASSOM) is basically a combination of a subspace method and a competitive selection and cooperative learning as in the traditional SOM [START_REF] Kohonen | Self-Organizing Maps[END_REF]. The single weight vectors at map units in SOM are replaced by sets of basis vectors that span some linear subspaces in ASSOM. A long-standing difficulty in the design of feature filters is the variation of input patterns due to typical transformations such as translation, rotation and scaling. By setting filters to correspond to pattern subspaces, some transformation groups can be taken into account automatically.

The input to an ASSOM network is called "episode", which is a sequence of pattern vectors that spans some linear subspace. This sequence is contructed by applying rotation, translation and scaling to the original local signatures. By learning the episode as a whole, ASSOM is able to capture the transformation kernels coded in the episode.

To construct these episodes, we focus our attention on regions of interest (ROI) in the images. Based on some psycho-visual experiments [START_REF] Hoffman | The role of visual attention in saccadic eye movements[END_REF], human vision system executes saccadic eye movements between salient locations to capture image content. Likewise, Tversky studies [START_REF] Tversky | Features of similarity[END_REF] showed that when we compare two images, we detect common and distinct concepts between these regions.

Our method tries to reproduce this extraction and distinction concept with a codebook learning strategy based on ASSOM algorithm. We firstly search salient locations in the images to be compared. Local visual features are then extracted from salient regions and projected onto a set of ASSOM-based learned visual prototypes, resulting in activation vectors.

Finally, we use these activation features to classify the image in adult or benign category with a NRBF neural network.

This method has been experimented for an adult content filtering method where the correct classification rate reaches 87.8%. The database is composed of 1, 110 adult images and 1, 200 benign images downloaded from Internet. The second category, known to be the rest of the world, is mainly constituted of landscapes, portraits and life scenes. This paper is organized as follows: In Section 2, we first present our image classification scheme based on ASSOM learning from ROI descriptions. Then, Section 3 contributes to some experimental results on the proposed classification schemes. And finally, conclusions are discussed in Section 4.

Image Classification Based on ASSOM Learning and Salient Regions of Interest

2.1 Multi-ASSOM Scheme (MAS).

As outlined in [START_REF] Duda | Pattern Classification[END_REF], a classification scheme is generally composed of three main steps : pre-processing, feature extraction and classification. In this paper, we mainly focus our attention on the two first items, the last being performed by a NRBF neural network. Our system architecture designs an ASSOM for each category, producing specific ASSOM units for different categories of image patches. This idea was explored in [START_REF] Zhang | Handwritten digit recognition by adaptivesubspace self-organizing map (assom)[END_REF] in the recognition of handwritten digits and produced promising results. But in their case, the image size is very small (25×20 pixels), permitting a direct learning through ASSOM. In their work, 10 ASSOMs are employed, one trained for each category of handwritten digits. For digit classification, a test digit is sent simultaneously to all the 10 ASSOMs, which output 10 error values. The ASSOM with the least reconstruction error determines the digit category. An obvious deficiency here is that there is no interaction between the different ASSOMs during the learning phase. An ASSOM learns the features of its own category, however it does not learn to distinguish features of other categories. The optimum decision surface is thus not guaranteed.

ASSOM 1 { } K k x k , , 1 , K ∈ Image patches ASSOM j ASSOM | | j max arg { } K k j k , , 1 , * K ∈               ) ( ) (
In our context, the dealt images have much larger sizes. So, we decide to use a local approach by extracting round image patches at salient locations. RGB patch informations are directly used to describe the local visual features.

Similar strategies have been developped in [START_REF] Csurka | Visual categorization with bags of keypoints[END_REF][START_REF] Quelhas | Modeling scenes with local descriptors and latent aspects[END_REF]. The principal differences are that the bag-of-keypoints representation is built from a K-means quantization and the classification is made by a Support Vector Machine(SVM).

Here, |J| ASSOMs are trained on these local descriptions, category by category. J denotes the number of ASSOM. For filtering use, two ASSOM neural networks are built for adult and benign classes.

To construct the feature vector y I for the final NRBF classification of the image I (See Figure 1), we operate as follows :

-For each patch x k , the |J| ASSOMs compete on it. The jth ASSOM produces an output xkj 2 defined by:

xkj 2 = max i∈Ij xkLi 2 , (1)
where I j is the set of indices of the modules in the jth ASSOM. In words, xkj 2 is the maximum value of the square of the orthogonal projection of the patch x k on the subspaces of the modules in the jth ASSOM. The j * k th ASSOM with the maximal output wins that patch:

j * k = arg max j∈{1,2,...,|J|} xkj 2 . (2)
-A counter n(j * k ) corresponding to this ASSOM network is accordingly increased by 1. When all the patches of the image I have been presented, the array of ASSOMs produce |J| counters of the patches won by the respective ASSOM. The feature vector is defined by:

y I = [n(1), . . . , n(j), . . . , n(|J|)] T , (3)
where the components are:

∀j ∈ {1, 2, . . . , |J|}, n(j) = k∈{1,2,...,K} δ(j * k , j) . (4)
δ(a, b) is the pulse function that takes the value 1 when a = b and 0 otherwise.

Wavelet-Based Salient Point Detection

According to the active vision mechanisms, the goal of salient point detectors is to find perceptually relevant image locations. Many detectors have been proposed in the literature [START_REF] Laurent | In depth analysis and evaluation of saliency-based color image indexing methods using wavelet salient features[END_REF][START_REF] Harris | A combined corner and edge detector[END_REF][START_REF] Bres | Detection of interest points for image indexation[END_REF]. The Harris detector [START_REF] Harris | A combined corner and edge detector[END_REF] aims at locating salient zones on corners by searching for the maxima of a function based on the local autocorrelation matrix of the signal. The detector in [START_REF] Bres | Detection of interest points for image indexation[END_REF] proposes to locate salient points in high contrasted area. The salient point detector in [START_REF] Laurent | In depth analysis and evaluation of saliency-based color image indexing methods using wavelet salient features[END_REF] uses a wavelet analysis to find pixels on sharp region boundaries. Working with wavelets in our previous work [START_REF] Laurent | In depth analysis and evaluation of saliency-based color image indexing methods using wavelet salient features[END_REF] is justified by the consideration of the human visual system for which multi-resolution, orientation and frequency analysis is of prime importance. In order to extract the salient points, a wavelet transform is firstly performed on the grayscale image. The obtained wavelet coefficients are represented by zerotrees as introduced by Shapiro [16]. This tree is then scanned at a first time from leaves to the root to compute the saliency value at each node. A second scanning occurs in order to determine the salient path from the root to the locations on the original image, where the raw salient points are located. The salient points are listed in order and a threshold τ , 0 < τ ≤ 1 is set to select the most salient points. By detecting salient points from luminance information only, the points located on boundaries of highlights or shadows are apt to be detected as salient. To remove false salient points caused by lumination conditions, a gradient image is built by using the color invariants proposed by Geusebroek et al. [START_REF] Geusebroek | Color invariance[END_REF].

This salient point detector reaches photometric invariance by combining the detection step with a recently proposed color invariance method [START_REF] Geusebroek | Color invariance[END_REF]. Experimental results in [START_REF] Laurent | In depth analysis and evaluation of saliency-based color image indexing methods using wavelet salient features[END_REF] show that the detected points are located on perceptually relevant image areas. Based on the detected salient points, the authors went further to design the salient signature that combines a color histogram with a texture measure. The proposed salient point detector and salient signature are applied to a contented-based retrieval system and the results are quite promising.

In this paper, we will combine the salient point detectors of our previous work with the ASSOM feature selector. Consequently, the ASSOM networks can be trained on small image patches centered on these salient points.

ASSOM Learning

As mentioned in the introduction, ASSOM is basically a combination of a subspace method and a competitive selection and cooperative learning as in the traditional SOM. ASSOM differs from other subspace methods by permitting to generate a set of topologically-ordered subspaces. That is to say, two units that are close in the map will represent two feature subspaces closed in the total feature space. In ASSOM, the unit is composed of several basic vectors that expand together a linear subspace. This unit is called "module" in an ASSOM neural network. This method aims to learn data features, without assuming any prior mathematical forms of their representation, such as Gabor or wavelet transforms, which are frequently encountered in the traditional image analysis and pattern recognition techniques [START_REF] Kohonen | Self-Organizing Maps[END_REF]. In other words, the forms of the filter functions are learned directly from the data.

The input to ASSOM is a group of vectors, called "episode". The vectors in each episode are supposed to be close up to affine transformations. There are mainly two phases in a learning process of ASSOM:

1. For an input episode, locate the winning subspace from ASSOM modules ; 2. Adjust the winning subspace and its neighbor modules in order to better represent the input episode.

For a linear subspace L of dimensionality H, one can find a set of basis vectors {b 1 , b 2 , . . . , b H }, such that every vector in L can be represented by a linear combination of these basis vectors. Such sets of basis vectors are not unique, however they are equivalent in the sense that they expand exactly the same subspace. For convenience of mathematical measures, the basis vectors are orthonormalized by the Gram-Schmidt process.

The orthogonal projection of an arbitrary vector x on the subspace L, notated as xL , is a linear combination of its orthogonal projections on the individual basis vectors, and can be computed by :

xL = H h=1 (x T b h )b h . (5)
If xL = x, then x belongs to L, else we can define the distance from x to L as xL = x -xL , by using the Euclidean norm. When several subspaces exist, the original space is separated from pattern zones and the decision surface between two subspaces, for example L 1 and L 2 , is determined by those vectors x such that xL1 = xL2 . By comparing the distances of a vector to all the subspaces, we can assign this vector to the nearest subspace.

In Kohonen's realization of ASSOM, the subspace is represented by a twolayered neural architecture, as in Figure 2. The neurons in the first layer take the orthogonal projections x T b h of the input vector x on the individual basis vectors b h . The second layer is composed of a single quadratic neuron and makes the output square sum from the first layer neuron.

The output of the whole neural module is then xL 2 , the square of the norm of the projection. It can be regarded as a measure of the degree of matching of the input vector x with the subspace L represented by the neural module. In the case of an episode, the distance should be calculated from the subspace of the vectors in the episode and that of the module, which are generally difficult to compute. Kohonen proposed another much easier but robust definition of subspace matching : the energy (sum of squares) of orthogonal input vector projections on the module subspace. Once the first phase occurred, the winning module with its neighbors adjust their subspaces to represent better the input subspace. A neighborhood function h (i) c is defined on the rectangular or hexagonal lattice (See Figure 3), where c notates the index of the winning module and i the index of an arbitrary module in the lattice. The neighborhood area defined by h (i) c shrinks with the learning step. Through this cooperative learning, the map will end at a topologicallyorganized status, where nearby modules have similar subspaces. The classical Kohonen's ASSOM learning algorithm works as follows: For the learning step t, 1. Feed the input episode x(s), s ∈ S, where S is the set of indices of vectors in the input episode. Locate the winning module indexed by c:

1 b 2 b H b Q x 1 T b x 2 ˆL x 2 T b x H b x T
c = arg max i∈I s∈S xLi (s) 2 , (6)
where I is the set of indices of the neural modules in the ASSOM. 2. For each module i in the neighborhood of c, including c itself, and for each input vector x(s), s ∈ S, adjust the subspace L i by updating the basis vectors b

(i)
h , according to the following procedure: (a) Rotate each basis vector according to:

b (i) h = P (i) c (x, t)b ′ (i) h . (7)
In this updating rule, b

h is the new basis vector after rotation and b

′ (i) h
the old one. P (i) c (x, t) is the rotation operator matrix, which is defined by:

P (i) c (x, t) = I + λ(t)h (i) c (t) x(s)x T (s) xLi (s) x(s) , ( 8
)
where I is the identity matrix, λ(t) a learning-rate factor that decreases with the learning step t. h hj ) max(0, |b

(i) hj | -ε), (9)
where ε is the amount of dissipation, chosen proportional to the magnitude of the correction of the basis vectors. (c) Orthonormalize the basis vectors in module i.

Experiments

In our experiment, the MAS scheme is applied to adult image filtering. There are respectively 733 adult images and 733 benign images in the training set of the image database. The test database is composed of 377 adult images and 467 benign images. The training of each ASSOM for each category takes T = 200, 000 epochs. The subspace dimension is set to H = 4 and the dimension of the ASSOM arrays is N = 10 × 10. The radius of a image patch is r = 14.5 pixels and thus the dimension of the input vector for the ASSOM arrays is 1, 971. These parameters are chosen by experimental results on the training set.

For our experiments, we configure our ASSOM network with the following rules to reach good learning results in terms of accurate input data representation [START_REF] Kohonen | Self-Organizing Maps[END_REF]:

λ(t) = T T +99t forms a monotonically decreasing sequence; h

(i) c (t) = 1, ||p c -p i || < µ(t) 0, otherwise.
The Euclidian norm is chosen and p i is the 2D location for the i th neuron in the network. µ(t) specifies the width of the neighborhood decreasing linearly during time t from √ 2

2 N to 0.5 . The classification performance on the training set is a true positive (TP) rate of 88.5% at a false positive (FP) rate of 13.2%. The TP rate is defined as the proportion of adult images that are correctly classified and the FP rate is the proportion of benign images that are incorrectly classified as adult. On the test set, the TP rate is 89.9% and the FP rate is 13.9%. The confusion matrix is presented in the Table 1. Some examples of correct and incorrect classification are then shown in Figure 4.

The performance is really promising, considering the value 0.9448 for the area under the curve (AUC), drawn in the Receiver Operating Characteristics (ROC) curve in Figure 5.

This multi-ASSOM architecture outperforms with 87.8% of correct classification rates a single ASSOM scheme (85.9%) and a single SOM scheme (78.35%) 1 .

Thus, we can see that the competition between an adult ASSOM and a benign ASSOM creates more precise feature vectors for NRBF classification. And the ASSOM algorithm permits to extract more robust features than the basic SOM method.

It is also very interesting to see the filters generated for the adult images and the benign images in Figure 6. We can observe that for the adult images, one of the basis vectors exhibit a orange color tone, and the other one shows obvious orientations. Thus, each final subspace tries to represent the data feature structure. That's why, when an adult image is fed into our scheme the adult ASSOM is stronger activated as shown in Figure 6.

Conclusion

In this paper, we proposed an original classification system using directly patch information. Based on the three main properties of ASSOM -which are dimension reduction, topology preservation and invariant feature emergenceour scheme filters images in a competitive way. This solution implemented for content-based image filtering gives us very promising results. A further improvement could be to distinguish model portraits from adult images, because this case composes chiefly the false detection. To get efficiency, the adult class training can be achieved with an object-based approach in order to skip background interferences. Furthermore, MPEG7 descriptors may certainly make the performances better than the simple RGB informations.

Fig. 1 .

 1 Fig. 1. The construction of the feature vector yI from patches of the image I in MAS. |J| ASSOMs compete on these patches and generate a sequence of winning ASSOM index j * k , k ∈ {1, 2, . . . , K}. n(•) counts the number of winning times for each ASSOM. The vector [n(1), . . . , n(j), . . . , n(|J|)] T forms the final feature yI, which is sent to the NRBF neural network

Fig. 2 .

 2 Fig. 2. Neural architecture of orthogonal projection of x on L

Fig. 3 .

 3 Fig. 3. Left: A rectangular topology. Right: A hexagonal topology. Each empty circle represents a neural module as shown in Fig. 2. The gray areas represent the neighborhood of the winning module indexed by c at a certain learning step.

c

  (t) is the neighborhood function defined on the ASSOM lattice with the support area shrinking with t. (b) Dissipate the components b (i) hj of the basis vectors b

Fig. 4 .

 4 Fig. 4. The left column presents correctly classified test images. The right column shows examples for misclassification.

Fig. 5 .Fig. 6 .

 56 Fig. 5. ROC curve of MAS on classification of adult and benign test images. Horizontal axis represents the FP rate and vertical axis the TP rate.

Table 1 .

 1 Confusion matrix of the MAS classification system with a ASSOM array of dimension 10 × 10. In this table, A=Adult, B=Benign.

	Classified as → A B
	A	339 38
	B	65 402

In our experiment, the best configuration for a single ASSOM scheme is : N = 10 × 10, H = 4, r = 14.5 ; and for a single SOM scheme : N =

× 20, r = 3.5.

⋆ This work was carried out during the tenure of a MUSCLE Internal fellowship.