WO2014009548A1

WO2014009548A1 - Method of formulating a multiclass discriminator

Info

Publication number: WO2014009548A1
Application number: PCT/EP2013/064829
Authority: WO
Inventors: Stéphane Canu; Jérôme Fournier; Yves GRANDVALET; Benjamin LABBE; Gaëlle LOOSLI; Benjamin QUOST; Liva RALAIVOLA
Original assignee: Thales; Institut National Des Sciences Appliquees De Rouen; Universite De Technologie De Compiegne; Centre National De La Recherche Scientifique; Universite D'aix-Marseille; Universite Blaise Pascal Clermont-Ferrand Ii
Priority date: 2012-07-13
Filing date: 2013-07-12
Publication date: 2014-01-16
Also published as: FR2993381B1; FR2993381A1

Abstract

The invention relates to a method of formulating a discriminator of objects which are obtained by an acquisition device furnished with a sensor, which comprises the following steps: - extraction of characteristics of n objects acquired, n > 1, the characteristics of an object constituting an observation x_i which is associated with a class, i varying from 1 to n, the discriminator comprising Q classes with Q ≥ 2, - construction of a decision rule for deciding between the classes. This step of constructing the decision rule comprises the following substep: determining for each class q a discrimination function representing this class, q varying from 1 to Q, on the basis of parameters ce, positive or zero which are defined by calculating (I) with M_ij = k(x_i,x_j)-λk(x_i,x_j) if x_i, and x_j belong to the same class or, M_ij = -λk(x_i,x_j) otherise, k being a predetermined function with positive kernel, and λ a predetermined real nonzero regularizing parameter, and with for each class q, (II), l_q being the set of indices of the observations of class q.

Description

METHOD FOR PRODUCING A MULTI-CLASS DISCRIMINATOR

The field of the invention is that of the automatic discrimination of observations, for example in the image monitoring or video surveillance systems where it is sought to decide from observations whether a situation is normal or not.

Discrimination is the assignment of an observation to a class among Q known classes of observations (for example, to assign an observation from an imaging system to the airplane or bird class or other ), while the term classification is reserved for the constitution of classes. Discrimination therefore occurs when classes have been previously defined and therefore associated with a label.

The problem posed, which is that of discrimination between different classes, becomes complex when classes are partially confused, or when only partial and / or noisy information is available on the observations to be discriminated, and even more so in the case of a lot of classes.

In the case for example of image monitoring systems, the variety of objects and backgrounds present in the natural scenes treated is very important and it is complex to discern the objects, especially as their distance and possibly their speed radial when these objects are mobile, are not known for acquisitions made in passive imaging. For example, at long distance, the boats can be very similar to airplanes (close radial velocities, almost uniform rectilinear motion, near intensity levels, etc.). The objects of interest must potentially be treated at a great distance, which augurs for low resolutions and therefore information that is not necessarily very rich in order to make a decision of discrimination. In addition, the shooting conditions (weather conditions, day / night conditions, reflections, glare, ...) change the signal on these objects, further complicating the task of discrimination.

We consider that an observation x is represented by characteristics (for example speed, SNR, intensity, shape signatures, etc.) that have been extracted from an object X. Recall in connection with Figure 1 the principles of automatic discrimination. There are two main phases:

- an off-line phase called the supervised learning phase during which the discrimination system is developed: the discrimination functions (one per class) that will be used to discriminate the observations, are defined as well as the assignment rules, starting from of a learning set composed of pairs (xi, yi), where xi is a learning observation (consisting of the characteristics extracted from the object Xi), yi the label of the known class to which this observation belongs, i varying from 1 to n, where n is the number of learning observations,

and an online phase during which an unknown object X _c will be recognized (in the discriminated sense). An image sensor for example forms the image of a scene from which an observation x _c is extracted consisting of the characteristics extracted from the object X _c or more generally a signal to be discriminated by the discrimination system. These characteristics are extracted by applying the same feature extraction method as for the off-line phase. The signal is then itself represented during a so-called representation phase which consists in calculating for each class the value of the discrimination function for this signal, then the class to which it belongs is determined at the end of a so-called assignment phase.

The difficulty is then to determine during this learning phase the boundaries between the classes, that is to say the decision rules making it possible to decide whether or not an observation belongs to a class, and to reject observations not belonging to any class defined a priori. The definition phase of these rules consists in partitioning the representation space thanks to borders; this phase consists in estimating boundary functions that minimize decision errors on a set of learning observations whose class is known, for a given definition of the characteristics of the object. The problem is therefore to find the right decision engine, in a multi-class context that is to say with a number Q of classes greater than or equal to 2. It is known to build these decision rules by relying on an SVM approach, an acronym for the term "Large Margin Separators" and also an acronym for "Support Vector Machines". This discriminative approach, described in particular in the document US Pat. No. 5,649,068 in the case of two classes, makes it possible to find, among all the separating hyperplanes of these two classes, the one which is optimal, that is to say which maximizes the margin between the two classes. hyperplane and samples (another name for learning observations). Vapnik has demonstrated [BE Boser, IM Guyon and VN Vapnik, A training algorithm for optimal margin classifiers. In Proceedings of the fifth annual workshop on Computational learning theory, pages 144-152. ACM, 1992] that the maximization of the margin corresponds to the minimization of the error of generalization of the decision on future observations for a finite set of learning observations. The principle of generalization is illustrated in Figure 2. The margin is the distance between the hyperplane and the closest observations; according to Vapnik's theory of learning, the optimal hyperplane is only characterized by the observations closest to it. These last called support vectors are therefore sufficient to define a fast online decision engine and maximizing the probability of a good decision. The contribution of each of these observations to the decision constitutes the solution of a quadratic program of known size, for which there are several resolution software solutions. Depending on the resolution solution chosen, the number of variables is equal to the number of learning observations n, or to the number of dimensions of the representation space d. The two modes of resolution are equivalent.

In order to be able to deal with the case of two non-linearly separable classes, this SVM approach is generalized by the introduction of variance variables and by the implicit projection of learning observations in a space of dimension greater than the space of the characteristics of the variables. observations, even of infinite dimension, in which the probability the existence of a linear boundary separating the learning observations of the two classes is greater. This is achieved through the use of a kernel function. The following functions are examples of core functions: polynomial functions, radial-based functions including Gaussian, quadratic rational, or hyperbolic tangent function.

Several methods have been proposed to extend the previous SVM solution to the multi-class case (Q> 2). We can mention the discriminative approach one against all, one against one, and the global approach.

The "one against all" approach consists in constructing Q binary discriminators by considering each class one after another, and each time assigning the +1 tag to the observations of the class considered and -1 to the observations of all other classes. This approach requires the resolution of Q quadratic problems of size n, where n is the total number of observations. This approach is illustrated in figure 3a for 3 classes, where one of the 3 binary discriminators is represented, in this case that of the class of circles against classes of squares and triangles.

The one-on-one approach is to build Q (Q-1) / 2 binary discriminators by comparing each of the Q classes to all the others individually. This approach requires the resolution of Q (Q-1) / 2 quadratic problems of size 2n / Q. This approach is illustrated in Figure 3b for 3 classes, where the two binary discriminators of the class of circles are represented.

The overall approach is to simultaneously build Q classifiers by directly solving a quadratic program of size nQ.

These three discriminative approaches do not allow for subsequent online use, to reject the distance intruders or anomalies (or "outliers" in English), ie to detect novelty, that is to say to reject the "remote" objects in the sense of a given metric of the learning data. This results in an increased risk of false alarms in cases of sensitive use as for monitoring systems.

We can also mention the generative approach, or contour (also called "One class-SVM" approach), in which we look for the outer limits of a class. This approach can be summed up by the formula "a class against nothing". It requires the resolution of Q quadratic problems of size n / Q. It also allows for subsequent online use, to reject intruders but does not take into account the superposition of classes which can lead to significant areas of rejection ambiguity (ie areas in which the decision is ambiguous between several classes). This approach is illustrated in figure 3c for 3 classes, where one of the 3 binary discriminators is represented, in this case the convex envelope of the class of the circles.

None of these methods presents a multi-class learning that is both efficient in computational complexity and discriminant. On the one hand, the complexity increases with the number Q of classes and the total number n of learning observations for the models "one-for-one", "one-against-all" and global. On the other hand, the "one-for-nothing" model does not inherently minimize decision ambiguities.

Consequently, it remains to this day a need for a method for constructing the decision rules of a multi-class SVM discriminator simultaneously satisfying all the aforementioned requirements, in terms of calculation complexity of the learning phase. , especially when the number of observations and classes is large (typically greater than 10 000 observations per 100 classes), distance rejection and ambiguous rejection and probability of error, when using the online discriminator .

The invention is based on an intermediate approach between the discriminative approach and the One class-SVM approach.

More specifically, the subject of the invention is a method for producing a discriminator of objects obtained by an acquisition device equipped with a sensor, which comprises the following steps:

extracting the characteristics of n acquired objects, n> 1, the characteristics of an object constituting an observation x, which is associated with a class, i varying from 1 to n, the discriminator comprising Q classes with Q> 2,

- building a decision rule between classes. It is mainly characterized in that this step of constructing the decision rule comprises the following sub-step: determining for each class q a discriminating function representing this class, q varying from 1 to Q, from ce parameters, positive or nulls that are defined by calculating

n n

m ^'n ^a i ΣΣ ^has JMI, j

α i = 1 j = 1

with:

M = k (Xj, Xj) -Ak (Xj, Xj) if x, and Xj belong to the same class or,

M = -Ak (Xi, Xj) otherwise,

where k is a predetermined positive kernel function,

λ a non-zero real parameter of predetermined regularization, and with for each class q, ^ α, = 1, l _q being the set of ie 1 _q

indices of observations of class q. According to a first embodiment, the discrimination function is a convex envelope g _q , the calculation defining the parameters ¾ then consisting in simultaneously minimizing the sum of the distances between all the pairs of convex envelopes and a weighted regularization term. a parameter δ with Δ = 1 / (Q + δ).

According to another embodiment, the discrimination function is a level curve f _q , the calculation defining the parameters ¾ then consisting in simultaneously minimizing the sum:

couplings between all the pairs of level curves, that is to say the cross scalar products of these functions, weighted by a parameter μ with λ = μ / (0.μ +1),

- a regularization term and

- the opposite of a bias p _q with f _q (Xi) - p _q ≥ 0.

In this way, the dimension of the quadratic program to solve to learn the decision boundaries is equal to n whatever the number Q of classes. This makes it possible to construct, in a computationally efficient manner, decision boundaries with a large number n of learning observations and a large number Q of classes. In addition, the use of convex envelopes or contour lines, will allow when using online to reject anomalies (distance rejection and / or ambiguity). When in addition we have, <C, C being a real number such as

C> where n _q is the number of observations of the class q,

gets a more robust solution. Since constructing the boundaries of a class takes into account the distribution of the observations in the other classes, it allows to take into account the possible superimposition of the classes. In this case and when the discrimination function is a level curve f _q , a positive difference variable μ is associated with each observation x, so that f _q (Xi) - p _q + μ, ≥ 0, and the calculation defining the parameters ce, then consists in simultaneously minimizing the sum:

- a regularization term,

- the opposite of a bias p _q

- and a weighted error term by C.

According to one characteristic of the invention, the default choice λ = 1 / Q promotes discrimination.

The regularization term may be the sum of the two standard squared discrimination functions or the sum of their norms one.

According to another characteristic of the invention, the step of constructing the decision rule comprises the following sub-step: determining allocation rules from said discrimination functions.

The invention also relates to a computer program comprising code instructions for performing the steps of the method as described above, when said program is executed on a computer. The invention also relates to a method for discriminating a candidate object obtained by an acquisition device equipped with a sensor, which comprises the following steps:

extraction of characteristics of this acquired object, according to the extraction of characteristics of the method as described above, the characteristics of this object constituting a candidate observation x _c,

. applying a decision rule to the candidate observation x _c which includes the following substeps:

calculating for x _c the values of the discrimination functions defined in the method as described above,

- Application of assignment rules determined in the process as described above.

The invention also relates to a computer program comprising code instructions for performing the steps of this method of discriminating a candidate object as described above, when said program is executed on a computer.

The objects are for example extracted from images acquired by an imaging device.

Other features and advantages of the invention will appear on reading the detailed description which follows, given by way of non-limiting example and with reference to the appended drawings in which:

FIG. 1 already described schematically represents the principles of automatic discrimination,

FIG. 2 already described represents schematically for two classes, several linear separations, among which that which maximizes the margin,

FIGS. 3 already described schematically illustrate three approaches for determining multi-class decision boundaries according to the state of the art: "one against all", "one against one", one-class,

FIG. 4 schematically illustrates the construction of decision rules for 3 classes, according to a first embodiment of the invention, Figure 5 schematically illustrates convex envelopes and contour lines for 3 classes, according to the invention.

From one figure to another, the same elements are identified by the same references.

During the (offline) learning phase, discrimination functions are developed on the one hand and assignment rules on the other hand.

We first focus on determining the functions of discrimination. According to the invention, we therefore seek for each class q a discrimination function representing this class, q varying from 1 to Q, from ce parameters, which are defined by calculating

n n

m ^'n ^a i ΣΣ ^has JMI, j

α i = 1 j = 1

with:

M = k (Xi, Xj) -Ak (Xi, Xj) if x, and Xj belong to the same class or,

M = -Ak (Xi, Xj) otherwise,

where k is a predetermined positive kernel function,

λ a real non-zero regularization parameter experimentally predetermined so as to favor discrimination with respect to rejection or to balance discrimination and rejection, as will be seen later,

and with for each class q ^ α, = 1, l _q being the set of ie 1 _q

indices of observations of class q.

The Applicant has demonstrated that this discrimination function can be equivalent, a convex envelope or a contour. This is detailed below.

We first consider the convex hull.

For two classes the equivalence between the search for a Vapnik separator hyperplane and the search for a representative of each of the classes within their respective convex envelopes has been demonstrated, these representatives being chosen so that the Interclass distance is minimal. This concept is here generalized to the case multi-class by considering for each class a convex envelope representing it while simultaneously minimizing the distances between all pairs of class representatives.

We therefore seek the convex envelopes g _q representing each of Q classes, qe [1, Q]. Among the n total observations, each class q is represented by a list of observations x ,, ie l _q , l _q being the set of indices of the observations of the class q. If n is the number of observations of class 1, n _q the number of observations of class q, n _Q the number of observations of class Q, on an ₌ card (li), .., n _q = card (l _q ), n _Q = card (l _Q ) with n + ... + n _q + ... + n _Q = n. An observation x, is a vector with dimensions (Xie R ^d ), d being the number of characteristics defined during the extraction of the characteristics of the object. The x, can be complex objects like graphs, probability distributions, automata, ... We put for all observations x: g _q (χ) = α, Κ (χ ,, χ) withΣα, = 1 and 0 <,, (1) ie l _q _{q q}

the parameters ¾ being the weights of the observations, that is to say a measure of the influence of each of the observations x, in the construction of the solution (so we have ¾ = 0 if x, is not a vector support) and k being a positive kernel function that associates a real value to any pair of observations x, x '. This positive kernel function can be, non-exhaustively:

- linear of the form k (x, x ') = x.x'

- polynomial of the form k (x, x ') = (x.x') ^d or (c + x.x ') ^d

- Gaussian of the form k (x, x ') = exp (- || χ-χ' || ² / 2σ ² ), where σ is the bandwidth of the function,

- sigmoid of the form k (x, x ') = tanh (a x.x' + b)

- laplacian of the form k (x, x ') = exp (- || x-x' || / o), where σ is the bandwidth of the function,

- quadratic rational of the form k (x, x ') = 1 / (1 + (|| χ-χ' || ² / σ))), σ being the bandwidth of the function. Determine for each class q a convex envelope g _q representing it, from ce parameters, which are defined by the following quadratic program solution:

n n

m ^'n ^a i ΣΣ ^has JMI, j

α i = 1 j = 1 with Σcc _j = 1, 0 <, and q varying from 1 to Q,

ie l _q

is to simultaneously minimize the sum:

- distances between all pairs of convex envelopes of the classes and

- a regularization term weighted by a parameter δ with A = 1 / (Q + δ). The parameter δ can be determined by minimizing a bound of the error or experimentally so as to maximize the discriminator's correct decision on a subset of randomly selected learning observations.

More precisely, it amounts to determining the parameters ¾ by solving:

with ^ α, = 1 and 0 <cci,

ie lq

H being the reproducing kernel Hilbert space induced by the chosen positive k kernel.

This is illustrated in FIG. 4 on which are represented 3 linearly separable classes and the decision boundaries F1 -2, F2-3, F3-1: class 1 with its convex envelope g-ι, class 2 with its convex envelope g ₂ class 3 with its convex envelope g ₃ , these envelopes being such that the distances | | Gi-g2 | | , |

| and 11 gi -Qsl | are simultaneously minimal.

The nature of the functions g _{q will} allow during the online phase to reject individuals from other classes and anomalies, that is to say, to ensure rejection ambiguity and distance. The term of weighted regularization makes it possible to relax the problem to be solved, that is to say to obtain an intermediate solution between the discriminative approach by level curve and the generative approach by convex envelope. In other words, it makes it possible to go from a problem of discrimination (δ = 0) to a generative type problem per level curve (one class-SVM when δ tends to infinity and λ = 0). The choice of δ makes it possible to balance the performance and rejection constraints. The problem is relaxed by modifying the weighting between the scalar products and the norms of the convex envelopes.

It is now considered that the discrimination function is a contour.

Determining for each class contoured f _q representative from this parameter, positive which are defined by solving the following quadratic program:

n n

m ^'n ^a i ΣΣ ^has JMI, j

α i = 1 j = 1

with ^ α, = 1, 0 <, and q varying from 1 to Q,

ie l _q

is to simultaneously minimize the sum:

- a regularization term,

- the opposite of a bias p _q . These biases are analogous to levels of probability; they are given explicitly by solving the problem (3) below or implicitly through the Lagrange multipliers of the equality constraints of the problem above,

couplings between all the pairs of level curves, that is to say the cross scalar products of these level curves, weighted by a real parameter μ with λ = μ / (Ο.μ +1) and μ ≠ - 1 / Q. The parameter μ which makes it possible to choose the importance of the term of coupling of the functions (by modifying the weighting between the scalar products and the norms of the functions), is determined experimentally so as to maximize the good decision of the discriminator on a subset of observations of learning.

More precisely, it amounts to determining the parameters ¾ by calculating: 1 Q .. .. n ^QQQ

f ^min Σ || ^f q || ⁺ -ΣΣ Σ ( ^f q ' ^f q') _H -Σ q

fq .Pq ^ q = 1 | _ | ^ q = 1q '= 1 q = 1 (3) the sign (.,.) _H denoting the dot product of the H space of Hilbert à Kernel

Reproducing, induced by the positive nucleus k chosen,

with: f _q (Xi) - p _q > 0 for all observations x, of class q and for all classes (q varies from 1 to Q).

The term of regularization makes it possible to relax the problem by explaining the degree of inter-function coupling, that is to say the couplings between the functions f _q and f _q >, by a modification of the weighting between the dot products and the Contour standards. This writing, which favors weak coupling between classes, makes it possible to better determine the decision boundaries, that is to say, to better discriminate classes, which will later allow the online phase to maximize the relevance of the discriminator.

The applicant has demonstrated that the level curves f _q are linearly associated with the convex envelopes in the following manner:

Q

fq (Xi) = gq (Xi) - ^ g _q (Xj)

q ^' = i Since the functions f _q and g _q are linearly linked, the membership tests of one candidate x, to one class rather than another, can therefore be done in an equivalent way on the functions f _q or g _q as shown in Figure 5 (we can also calculate g and f). Indeed we have: arg max f _n (x) = arg max g _n (x)

y qE [l, Q] ^q 'qE [l, Q] ^yq '

In this way, the complexity of calculating the learning of convex envelopes g _q or contour lines f _q , is that of a quadratic program with n unknowns whatever the number Q of classes. This makes it possible to construct, in a computationally efficient way, decision boundaries with a large number of observations and a large number of classes. Moreover, since the construction of the boundaries of a class takes into account the distribution of observations in the other classes, this allows the interaction between the classes to be taken into account.

We also have λ = μ / (Ο.μ +1) = 1 / (Q + δ). When δ = 0 (or μ very large) we have λ = 1 / Q, as we can see in the table of variation of μ and λ (non-zero) as a function of δ below, which means that the discriminator will ensure during the online phase that the function of discrimination but not rejection distance or ambiguity. As soon as δ> 0, the discriminator will insure discrimination and rejection in distance and in ambiguity.

Whether it is the convex hull or contour approach, these discrimination functions are determined in particular according to a regularization term defined from the standard of the discrimination functions considered. According to an alternative, this standard can be replaced by a pseudo-norm.

To treat the case of nonlinearly separable classes, we introduce an additional condition: ¾ <C, C being a real number. The only constraint on the value of this upper bound C is not to prevent the equality constraints from being satisfied, which imposes:

C≥max _qE [ _1iQ ] -i-.

nq This does not introduce a change for the calculation of convex envelopes. On the other hand, the contour lines are then obtained thanks to the introduction of positive difference variables μ, associated with each observation so that f _q (Xj) - p _q + μ, ≥ 0 for all the observations x, of the class q and this for all classes (q varies from 1 to Q) and by adding to the objective function to be minimized of equation (3) an error term (the sum of these variance variables) weighted by the parameter C.

The focus is now on the assignment rules. Several rules can be defined. For example, the following assignment rules can be adopted.

A candidate observation (whose characteristics have previously been extracted) is assigned to:

- a class q if it is the only one to respond positively,

- a rejection of ambiguity as soon as more than one class responds positively,

- distance rejection if no class responds positively (= all classes respond negatively).

Or again, a rejection in distance is decided when the maximum of the scores of the functions g _q or f _q is lower than a threshold null or determined experimentally. The rejection of ambiguity is decided when there is no distance rejection and the difference between the two best scores is lower than another zero or experimentally determined threshold.

During a step of the online phase called the representation step, the discriminator calculates for a candidate observation x _c , the values of the discrimination functions for all the classes, ie gi (Xc), g _q ( _xc ), go (x _c ) if the convex hull approach was chosen or fi (Xc), fq (x _c ), fo (c) if the contour approach was chosen. We then apply in a so-called assignment step, the assignment rule at the end of which, the candidate observation x _c is assigned to a class q (because we have _q (x _c )> 0 and g _q ( x _c ) ≤ 0 for q '≠ q, using the first assignment rule example mentioned above) or rejected remotely or ambiguously.

These are the steps of the decision rule. A discriminator has been presented in a standby or imaging surveillance context. More generally, it can also be applied to automatic discrimination in a surveillance system of objects acquired by means of a sensor that it is a sensor of an imaging device, or a sound or radar device or any other sensor.

The present invention can be implemented from hardware and / or software elements. It may be available as a computer program product on a computer readable medium. The support can be electronic, magnetic, optical, electromagnetic or be an infrared type of diffusion medium. Such supports are, for example, Random Access Memory RAMs (ROMs), magnetic or optical tapes, disks or disks (Compact Disk - Read Only Memory (CD-ROM), Compact Disk - Read / Write (CD-R / W) and DVD).

Claims

Process for producing a discriminator of objects obtained by an acquisition device equipped with a sensor, which comprises the following steps:

for extracting characteristics of n acquired objects, n> 1, the characteristics of an object constituting an observation x, which is associated with a class, i varying from 1 to n, the discriminator comprising Q classes with Q> 2,

constructing a decision rule between classes,

characterized in that said step of constructing the decision rule comprises the following substep: determining for each class q a discriminating function representing that class, q varying from 1 to Q, from positive or null parameters qui which are defined by calculating

n n

mi ⁿ ΣΣ ^a i ^a jMi, j

α i = 1 j = 1

with M = k (Xi, Xj) -Ak (Xi, Xj) if x, and Xj belong to the same class or, My = -Ak (Xi, Xj) otherwise, k being a predetermined positive kernel function, and λ a nonzero real parameter of predetermined regularization, and with for each class q, Σccj = 1, l _q being the set of

ie l _q

indices of observations of class q.

Process for producing a discriminator according to the preceding claim, characterized in that the discrimination function is a convex envelope g _q , the calculation defining the parameters ce, then consisting in simultaneously minimizing the sum of the distances between all the pairs of convex envelopes and a regularization term weighted by a parameter δ with A = 1 / (Q + δ).

Process for producing a discriminator according to the preceding claim, characterized in that 0 <cci <C, C being a number real as C> max _qe [ _{1 Q} ] -, n _q being the number of observations of the nq

class q.

Process for producing a discriminator according to claim 1 or 2, characterized in that the discriminating function is a level curve f _q , the calculation defining the parameters ce, which then consists in simultaneously minimizing the sum:

- a regularization term and

- the opposite of a bias p _q .

A method of making a discriminator according to the preceding claim, characterized in that 0 <cci <C, where C is a real number such that C> max _qe _{[1 Q]} -, n being the number _q of observations of the

nq

class q, in that a variable of positive difference μ, is associated with each observation x, so that f _q (Xi) - p _q + μ, ≥ 0 and in that the sum to be minimized further comprises a C-weighted error term

6. Process for producing a discriminator according to one of claims 2 to 5, characterized in that λ = 1 / Q. A method of constructing a discriminator according to one of claims 2 to 6, characterized in that the regularization term is the sum of the two standard squared discrimination functions or the sum of their norms one.

8. Process for producing a discriminator according to one of the preceding claims, characterized in that the step of constructing the decision rule comprises the following substep: determining assignment rules from said discriminating functions.

9. Process for producing a discriminator according to one of the preceding claims, characterized in that the objects are extracted from images acquired by an imaging device.

A computer program comprising code instructions for performing the steps of the method of any one of claims 1 to 9 when said program is run on a computer.

1 1. A method of discriminating a candidate object obtained by an acquisition device provided with a sensor, which comprises the following steps:

extraction of characteristics of this acquired object, according to the extraction of characteristics of the process for producing a discriminator according to one of claims 1 to 9, the characteristics of this object constituting a candidate observation x _c ,. application of a decision rule to observation x _c which includes the following substeps:

calculating for x _c values of the predefined discriminating functions in the process for producing a discriminator,

- application of predefined assignment rules in the process of developing a discriminator.

A computer program comprising code instructions for performing the steps of the method of claim 11 when said program is run on a computer.