Películas experimentales

3.2.1

Levels of Fusion

According to [132] (and references herein), biometric systems can be combined at several architectural levels, as follow:

• sensor, e.g., weighted sum and concatenation of raw data, • feature, e.g., weighted sum and concatenation of features,

• score, e.g., weighted sum, weighted product, and post-classifiers (the conventional machine-learning algorithms such as SVMs, MLPs, GMMs and Decision Trees/Forests); and

• decision, e.g., majority vote, Borda count, Behavioral Knowledge Space [138], Bayes fusion [74], AND and OR.

The first two levels are called pre-mapping whereas the last two levels are called post-mapping. Algo- rithms working in-between the two mappings are called midst-mapping [132]. We are concerned with the

score level fusion (hence post-mapping) in this thesis. Note that we do not work on the decision level

fusion but the score level fusion because much richer information is available at the score level, e.g., user- specific score statistics. In fact, an experimental study in [74] shows that the decision level fusion does not generalize as well as the score level fusion (although this was the objective of the paper).

3.2.2

Decision Functions

Let us denote C (for client) and I (for impostor) as the two class labels the variable k can take, i.e., k _{∈ {C, I}. Note that class C is also referred to as the genuine class. We consider a “person” as a} composite of data for various biometric modalities, which can be captured by biometric devices/sensors, i.e.,

person=_{tf ace, tspeech, tf ingerprint, . . .},

where tiis the raw data, i.e., 1D, 2D and multi-dimensional signals, of thei-th biometric modality.

To decide whether to accept or reject an access requested by a person, one can evaluate the posterior

probability ratio in logarithmic domain (called log-posterior ratio, LPR):

LPR _{≡ log}µ P (C|person) P (I_|person)

¶

= logµ p(person|C)P(C) p(person_{|I)P (I)}

¶ , = logp(person|C) p(person_|I) | {z } + logP (C) P (I) | {z } , = logp(person|C) p(person_|I) − log

P (I) P (C) ≡ y

llr

− ∆, (3.1)

where we introduced the termyllr _{– also called a Log-Likelihood Ratio (LLR) score – and a threshold}

∆_{≡ logP(C)}P(I) to handle the case of different priors. This constant also reflects the different costs of false acceptance and false rejection. In both cases, the threshold∆ has to be fixed a priori. The decision of accepting or rejecting an access is then:

decision(LPR) = ½ accept if LPR> 0 reject otherwise, (3.2) or decision∆(yllr) = ½ accept ifyllr _{> ∆} reject otherwise, (3.3)

3.2. NOTATIONS AND DEFINITIONS 23

Letyprob _{be the probability of being a client, i.e.,y}prob _{≡ P (C|person) and using the definition of}

LPR _{≡ log}³P(C|_P(I|personperson₎)´, the decision function of (3.2) can be written asP (C_{|person) > P (I|person)} orP (C_{|person) > 0.5, since P (C|person) + P (I|person) = 1. In terms of y}prob_{, this decision function is:}

decision∆(yprob) =

½

accept ifyprob_{> 0.5}

reject otherwise, (3.4)

Note that the prior probability has already been absorbed, i.e,P (C|person) ∝ p(person|C)p(C).

We callyllr_{an LLR score whereasy}prob _{a probability}1_{. In theory, the decision functions of (3.3) and}

(3.4) are equivalent because both can be derived from (3.2). However, in practice, the explicit presence of a threshold in (3.3) is more convenient because the prior probabilities (P (C) and P (I)) can be adjusted

separately from the LLR score. For this reason, (3.3) is more commonly used in the literature. For the rest

of the discussion, we will writey ≡ yllr _{so that we consistently use LLR in our discussion unless stated}

otherwise.

3.2.3

Different Contexts of Fusion

From an architectural view point, the (LLR) scorey can be explicitly written as:

y_{≡ f}θ(fe(s(t))), (3.5)

where,s is a sensor capturing a particular biometric trait t, feis a feature extractor,θ is a set of classifier

parameters associated to the classifierfθ. We also denote x≡ fe(s(t)) when only the extracted features

are concerned.

When considering different fusion contexts, the scorey is associated to a subscript i, which takes on a different meaning. The score can be summarized as follows:

yi(person) =        fθ(fe(s(t[i]))) if multi-sample fθ(fe(si(ti))) if multi-modal fθ(fe,i(s(t))) if multi-feature fθ,i(fe(s(t))) if multi-classifier, (3.6)

where t denotes any given one of the tibiometric traits fori∈ {face, speech, . . .}, t[i] denotes the i-th

instance (in time) of the biometric trait t, and tidenotes thei-th biometric trait. As in common biometric

applications, we assume that a dedicated sensor is designed to capture a specific biometric trait, i.e.,si(ti).

Note that the indexi takes on a different meaning in any of the four contexts in (3.6). For example, i denotes thei-th instance in the multi-sample case, the i-th biometric modality in the multi-modal case, the i-th feature set in the multi-feature case, and the i-th classifier in the multi-classifier case.

To simplify the notation, we write yi instead ofyi(person), while bearing in mind that yi is always

dependent on the “person” (in the sense of composite 1D or 2D signals as captured by biometric devices) who makes an access request. Without loss of generality, we assume that for each access request, there are yi|i ∈ {1, . . . , N} scores available. We further write y to refer to the output of any of the arbitrary systems

i∈ {1, . . . , N}.

Let y = [y1, . . . , yN]′ be the vector of system outputs to be combined. To decide if an access should

be granted or not, a fusion classifier fCOM : RN → R must be defined. This can be expressed by

yCOM = fCOM(y). Note that the decision function in (3.3) can still be used for the score yCOM. The

different types of fusion classifiers of the formfCOM will be discussed in Section 3.4. In the next Section

we will examine different score types commonly used in the literature.

1_{There is an increase use of y}prob′_{= P (C|y) in fusion, e.g., [60], where y is an output score and P (C|y) is considered a score-}

normalization procedure intended to approximate the ideal probability yprob_{= P (C|t) and t is a biometric trait. While y}prob_{is a}

true probability, yprob′_{can, at best, be the score-level approximation of y}prob_{. No distinction is made between y}prob_{and y}prob′_in

24 CHAPTER 3. SCORE-LEVEL FUSION