Performance Measures for Biometric Recognition

1. Introduction

This paper extends the elementary intra-characteristic derivation of biometric failure rates to the case where the pseudo-statistical properties of an ensemble of biometric features from different biometric subjects are considered. While the intra-characteristic consideration only treats the stochastical properties of single comparisons between sample pairs of the same or two different characteristics, the inter-characteristic analysis extends the definitions to all pairs of different characteristics within the ensemble. This greatly helps to understand the fact that different pairs of features show quite different properties in real life. The main targets are to prepare the basics for optimum biometric recognition procedures and the cost-efficient evaluation of their performance.

2. Biometric Features and Metric Spaces

A typical biometric recognition system can be separated into four function blocks:

• Capturing using an electronic sensor
• Modality detection
• Feature extraction
• Feature comparison with reference features

During capturing a sample of the biometric characteristic is recorded which must "contain" the biometric feature. The Modality Detection is responsible for the availability of the desired modality (finger, face, iris, etc.) and may refuse images which do not meet predefined quality and other requirements, such as liveness. During feature extraction those information is omitted as far as possible which does not contribute to the requirements of uniqueness, universality, and permanence. The extracted feature is the basis for comparison. It can be expressed mathematically as a multi-dimensional vector or at least as a list of property parameters.

The main task of feature extraction can alternatively be described as a method to reduce dimensionality. For example, many biometric systems start with capturing a 2D image of the biometric characteristic using a digital black & white camera. If the image sensor has N x M pixels, ideally 1 dimension per pixel, i.e., NM dimensions, can be considered, where each dimension typically may represent 256 gray values. If the image represents a fingerprint, the dimension will reduce, e.g., to the number of extracted minutia multiplied by the number of minutia properties. The number of minutia properties is typically 4: horizontal position, vertical position, direction, and type (ending or bifurcation).

An important measure to reduce dimensionality is alignment. Alignment can be expressed as a projection operator with the property that already aligned data will not change if alignment is repeated. For example, the most common 1-dimensional feature of human is height. In a 3-dimensional Euclidean space, an advantageous alignment is the projection of the feet to the coordinate (x1,y1,z1)=(0,0,0) and the top of the head to (x2,y2,z2)=(0,0,h) where h is the height of the person. In this case, the coordinate z bears all information required. This alignment can be attained in at least two ways: Either the person is asked to stand upright at a certain point or the coordinates of feet and head are measured and then projected to (0,0,h) by using two translations and one rotation. This way the number of relevant dimensions reduces from 6 to 1. This example also demonstrates that alignment is not only a matter of feature extraction but also of the capturing process. In cases where alignment is not completely possible, even the comparison function is affected. For example, while iris images can simply be aligned in horizontal and vertical directions, rotation cannot be compensated for, especially if only one eye is captured. In such cases, the comparison stage has to try all rotations until the "distance" between the features approaches a minimum.

Sometimes, the camera itself delivers unwanted projections. For example, faces are embedded in the 3 dimensional Euclidean space. If the image is taken by a 2D camera, certain rotations are difficult to compensate, if the face is not aligned mechanically in front of the camera. Using 3D cameras, such unintentional rotations can be simply compensated in the digital domain. Additionally, the features to be extracted are hardly affected by illumination in the 3D case [3DFaceProject].

For fingerprints, alignment in the third direction is done mechanically by pressing the finger on the 2D sensor surface. During feature extraction, alignment is more difficult. Horizontal and vertical alignment is possible, if the fingerprint pattern would show a common property for orientation. Such a property could be the core of the ridge pattern flow. Unfortunately, small sensors may not show the center at all and many fingerprints show more than 1 center. But even if a core is found, this only allows translational alignment. Rotations remain a problem which can be solved by finding the best orientation during comparison, e.g., by trial and error. This will, however, increase computation time significantly.

Beside their distinctive properties, most biometric characteristics also show common properties which can be used to detect their presence after capturing. Such properties are only distinctive between different biometric modalities, not between different characteristics of the same modality. Although these properties cannot be used for recognition of individuals they are advantageous especially for alignment. Examples are the ridge structure of fingerprint, the presence of eyes, nose and mouth in faces, the general outline of a hand with five fingers and the special shape of an eye with pupil, iris and surrounding. Ultimately, detection and recognition use the same mathematical procedures. In this context, we will understand Modality Detection as recognition based on known common properties with respect to different individuals of a biometric characteristic (which define the modality), whereas Feature Detection is based on the known distinct properties with respect to different individuals of the same biometric characteristic. If said common properties are not present, no feature processing need follow, and the recognition fails anyway.

Since property detection and property estimation are mathematically similar tasks, detection may be viewed as an add-on of estimation. (Usually, detection requires that some measure of the parameters exceeds a certain threshold.) Generally, it must be regarded that the functional blocks capturing, detection, extraction, and comparison cannot always be separated exactly.

2.1 Feature Spaces

Here we understand an individual to be separable, not necessarily to be unique as this is defined, e.g., in [Wikipedia]. Separability is guaranteed by the subject's virtual index only. This index is not measurable. For that reason we need measurable attributes to distinguish individuals practically.

The pairing (sn, an) may be of different strength and of different origin. Generally, individuality may be awarded artificially (password, RFID, clothes) or may be a naturally fixed part of the subject's body (face, iris, fingerprints).

As already mentioned, the space A itself may be a Cartesian product of elementary attribute spaces. This allows us to describe an attribute an as vector with arbitrary many components, e.g.,

an = (color, height, name, history, ...) (2.5)

helping us to create more combinations so that we have enough different attributes an to distinguish all individuals in ℑ. 

On the other hand, attributes which are common for a non-empty set of subjects, may be of interest. If ℑ is the set of all individuals, e.g., all living humans, then a population is a subset ℘ (P script) of ℑ which is united by certain common properties of its members. Such properties could be profession, gender, race, hobby, name, etc. Of special interest are populations which have a measurable influence on biometric recognition. Regarding non-idealities, special populations may show better or worse failure rates, such as the population of bricklayers which may have problems with fingerprint recognition. When determining failure rates for biometric recognition systems, the properties of a population are essential when comparing the rates between different applications.

Common properties can also be used to define biometric modalities like behavior type, face geometrics, fingerprint pattern, iris pattern, or any other property like knowledge (passwords), possession (smartcard, key, wear), or marks like scars and tattoos. Here the focus is on common properties: What is common to all faces or to all fingerprints or to all passwords? The question, "What distinguishes all password from all faces or from noise?"  can be used for modality detection to decide whether the desired biometric (or other) characteristic has been measured or not. For modality detection, the distinctive properties of a characteristic are irrelevant.

The distinctive properties required for recognition are collected as feature vector fn. As feature space F we define a set of those attributes fn which can be used for comparison with other members of F. What can be compared is only depending on the capabilities of the comparison stage in Figure 2.1. Consequently, the feature space F is a subspace of all attributes and their combinations A:

F ⊂ A (2.6)

To prevent loss of generality, we also allow raw signals, coming directly from the sensor, to be compared by an appropriate comparison stage and to define a generalized feature space. The raw or sample signals create the sample space Σ. However, using Σ as generalized feature space may not necessarily help vividness. The intention is to allow the comparison function to provide optimum processing in a mathematical sense, at least theoretically.

In practice, not enough a priori knowledge about the biometric characteristic and the inherent character of the noise which deteriorates the ideal biometric feature may be available to define the optimum algorithm or the problem is too complicated to result in a simple solution. For that reason, most practical systems follow the block diagram of Figure 2.1. If so, the feature space contains all biometric features which come from a biometric characteristic or are stored as biometric references, resp. templates. Each feature fn as an element of F may comprise subfeatures (more elementary properties) or feature components. Nevertheless, if we talk about features as plural, we don't mean the subfeatures of one feature vector but different elements of F. We note that F need not be a vector space. Generally,

|S| ≠ |F| (2.7)

More precisely, we regard F as the space of all possible features within a specific comparison-oriented representation. That is, if the comparison block of Figure 2.1 cannot process a feature, the feature cannot be a member of F.

From F the subset F of all used  features may be separated. "Used" means, each feature is used by at least one subject from S.  In the ideal case, for each known subject sn ∈ S there is exactly one feature fn and a bijective map ⓘn ∈ ℑ : sn ↔ fn called individual. In this case

|S| = |F| (2.8)

If features are noisy, F becomes a random variable with usually different realizations F(ω) for different ω. In this case we get

|S| ≥ |F(ω)| (2.9)

F(ω) ⊂ F for every ω ∈ Ω (2.10)

since different features may accidentally fall together. If individuals are not unique even in the absence of noise, we may even get

|S| > |F| (2.11)

For example, purely genotypic features may be equal for monozygotic twins, making them indistinguishable.

The case |S| < |F| would mean that there are more used features than subjects. This will be excluded by the requirement that at most one feature from F is assigned to one subject and is equivalent to the definition of functions [Wikipedia].

Example 2.1: Human height embedded in three dimensions Each individual is characterized by two points: the top of the head and the bottom of the feet. In the 3D Euclidean space this requires two points with three coordinates each, resulting in 6 numbers per individual. If we assume no quantization in the lengths, F comprises infinite many 6-tuples, which can be compared using a suitable comparison algorithm. Alternatively, an alignment can be performed by preprocessing as indicated in Figure 2.1. Then F will be the space of all human heights, which can be compared with each other by very simple methods (metrics).

Example 2.2: Human height, aligned If the individuals are already aligned in 3D space, one individual can be characterized by a single number, the distance between the two 3D points given in Example 2.1. This can be done by direct measurement or by calculating the Euclidean distance between the two points given in Example 2.1. In this example, alignment is performed as part of the capturing process, while in the previous one, alignment is done after capturing, either by the comparison stage or by the preprocessing steps.

Example 2.3: Fingerprint images If fingerprints are compared without preprocessing, the raw images form the feature space F. Its dimension is given by the number of pixels. The range of each pixel is typically given by the size of one Byte, e.g., 0...255. The number of elements is easily computed, e.g., as |F| = L x M x 255 where L and M are the number of pixels in horizontal and vertical direction. In comparison to other representations, this one is computationally more intensive. One reason for this fact is that the reference must also be stored as image. This may require numerous computation steps to be repeated for each new comparison.

Example 2.4: Fingerprint minutiae Since it is well known that the minutiae patterns are the carrier of uniqueness, most systems only store the minutiae properties as reference. This requires preprocessing to extract the minutiae from the fingerprint image. Remaining is a list of a variable number of minutiae which are each described by their horizontal and vertical position, the minutia type (ridge bifurcation or ending), and the angle of the ridge structure at the location of a minutia. The dimension of F is 4K if K is the maximum number of minutiae of one feature. While position and angle are expressed as a quantized real number, the minutia type only requires 1 bit. In this case, the feature components need not have the same structure.

Example 2.5: Hair color Though hair color of an individual may not be a very distinctive feature, it may nonetheless serve as a good example. This feature is by far not perfect. It may change over time, can simply be altered artificially, is not available for all individuals, may change locally etc. The feature space may be a set of hair colors such as F ≔ {brown, black, blond, auburn, red, white}. More technically, F may be defined as the space of all optical hair spectra or more practically as a 3-component color vector.

Example 2.6: Keystroke For keystroke recognition, the raw information are the timing components (and sometimes also the pressure) of certain key successions. The resulting features, together with their stochastical properties, are very distinctive between different persons.

2.1.1 Combination of Features

Unfortunately, real-life situations are more complicated. For example, a fingerprint feature may have continuous and binary components. The location of a minutia is such a continuous component which may easily be treated as vector space component. On the other hand, minutia type is a binary feature component which requires different treatment. Often, certain minutiae are not available. In this case it makes no sense to replace them by a null vector since then averaging would deliver useless results.

Furthermore, features should be aligned before averaging. Practically, this can be handled by aligning two features by calculating the minimum distance for all possible alignments. If this minimum is zero or below a threshold, it is indicated that both features come from the same biometric characteristic. If so, a mean value may be taken.

Example 2.7: Binary features and median Let F be a feature space such that F = {0,1}. Let fn ∈ F(ωn) be a random feature with realizations f1 = 0, f2 = 0, f3 = 1, f4 = 0, f5 = 1 (2.15) The median [Wikipedia] value is seen to be median{f1, f2 , f3, f4, f5} = median{0, 0, 1, 0, 1} = median{0, 0, 0, 1, 1} = 0 (2.16) That is, the value with the higher frequency is taken. Note: If we have an even number of elements to be combined, the median is not unique as the number of 0s and 1s may be equal. In such a case, any value (0 or 1) may be taken as result. Statistically this should have little impact. (Sometimes, for the non-unique case, ½ is defined as median. We avoid this since ½ is not an element of F and thus may confuse the biometric comparison operation).

Example 2.8: Adding a unique ID as "feature component" If biometric features are not sufficiently unique (this may also hold for passwords), some unification may be advised. This can most favorably be attained by adding a unique feature component to the other components of a biometric feature. The trick is that the unique component may be public while the non-unique one may represent a secret or something that can only be delivered by the dedicated individual. (Otherwise the unique component would be sufficient!) For passwords this method is very common since the very beginning of its use by combining the password with a user ID which can be measured easily with little errors. A user ID may also be a solution for biometric features since user IDs can be more rapidly be identified than biometric features. It is important to use an appropriate distance measure for the new combination to get all the advantages one expects. Otherwise, the unification may fizzle out as we shall see in Example 5.5.

2.2 Distance measurements

Depending on which conditions hold, different definitions can be given [Wikipedia]:
 

Which distance measure is taken depends on the concrete realization. For modality detection, the feature space F should be replaced by a suitable subspace of A. The distance measure over this subspace may be completely different from that over F.

Example 2.9: Human height: Euclidean metric After having reduced the dimensions to 1, a suitable distance measure for measuring the difference of two heights f and g is the Euclidean metric d(f,g) ≔ |f − g| where f, g ∈ ℝ and D = ℝ⊕.

Example 2.10: Discrete metric Let F be the space of all PINs including zero: F = ℕ0 ≡ ℕ⋃{0}. Then the distance measure δ(fi,fj) ≔  { 0 if fi = fj 1 if fi ≠ fj (2.17) is a metric called discrete metric δ: F x F → D = {0,1}

Example 2.11: Correlation coefficient Let F be a Hilbert space [Wikipedia] with scalar product <•,•> and norm ‖•‖= √<•,•>. Then the correlation coefficient χ(f,g) can be defined as  χ(f,g) ≔  |<f,g>| ‖f‖ ‖g‖ (2.18) for all nonzero f, g ∈ F. χ is a measure for the similarity of two vectors f and g which is independent of the amplitude since χ(αf,βg) = χ(f,g). χ can be written as χ(f,g) = 1 - ½ ‖ f ‖f‖ − g ‖g‖ ‖² (2.19) If f = αg for any α≠0, then χ(f,αg) = 1 which represents the maximum due to Schwartz's inequality. In the case of minimum similarity we get χ(f,g) = 0 which is reached if f is a symmetric vector and g an antisymmetric one. Since similarity intuitively behaves inversely to distance we define a distance measure as d(f,g) ≔ ‖ f ‖f‖ − g ‖g‖ ‖ (2.20) This measure is only a pseudometric since DIS3 is not fulfilled, e.g., in the case g ≔ αf ≠ f where 0≠α∈ℝ, we get d(f,g) = 0.

2.2.1 Composite distance measures

The weighting factors α and β may have influence on D. For α > β, the ordering of the corresponding distance values in a discrete D is different from α < β. If α = β, this may reduce |D|. Both has a strong influence on the failure rates to be defined as we shall see!

Composite distance measures may be required when a feature comprises feature components of different type, for example real components such as length and binary components such as "available" and "missing".

Example 2.12: Hamming metric The Hamming metric is well known as Hamming distance [Wikipedia] and describes the number of different symbols when comparing two strings. Since the equality of two symbols can be measured by the discrete metric δ (Example 2.10), we can describe the Hamming metric as composite metric using the discrete metric. Let F be a feature space with N different feature components. Each feature component is characterized by one symbol (which may be a number or a character etc.) and is represented by a feature subspace Fn such that F = F1 x F2 x F3 x ... x FN (2.22) Then the Hamming metric d can be written as d(f,g) = δ(f1,g1) + δ(f2,g2) + δ(f3,g3) + ... + δ(fN,gN) (2.23) where f = (f1, f2, f3, ... , fN) ∈ F, g ∈ F, and fn ∈ Fn. δ is the discrete metric (Example 2.10).

3. Biometric Recognition in Metric Spaces

3.1 Recognition

1. the features are equal and belong to the same characteristic or
2. the features are unequal and thus belong to different characteristics.

Example 3.2: Human height: Absolute Values Fig. 3.1: Biometric height features as points in a 1D feature space (F = ℝ⊕)

One of the most interesting properties of the space of features are the distributions of distances between the features. To build a statistics one may start with a certain feature and determine the distances to its neighbor features. In an N dimensional feature space, all feature points within an N-dimensional ball around the actual feature are to be counted. With increasing radius of the ball, the number of features included may increase faster or slower, depending on their local distribution. This is similar to stars in universe. Stars may be crowded in galaxies or the space may be nearly empty over large distances.

Example 3.3: Human height: Distances from a gregarious feature When the one-dimensional (1D) feature space F contains N features, 2N distances (with N−1 non-trivial distances) from one feature to all other features can be calculated. (Non-trivial means: excluding the zero distance of a feature to itself and assuming the two back and forth distance values between two different features being equal according to DIS4.) Depending on which feature is selected as point of origin, different distance distributions may result. In this example, a feature amid an aggregation of features is chosen, thus delivering a lot of small distance values. Figure 3.2 shows the distribution which represents points in the distance space D ⊂ ℝ⊕. Fig. 3.2: Distance distribution of the height features Fig. 3.1 from a gregarious feature

Example 3.4: Human height: Distances from a lonely feature If we start with the same feature space F as in Example 3.3, but choose a lonely feature as origin, a completely different distribution of distance values may result which are far from zero. This is an expression of the fact that lonely features are easier to separate than features within a crowd and confirms the practical observation that different features show different recognition failure behavior. Fig. 3.3: Distance distribution of the height features Fig. 3.1 from a lonely feature

Intuitively, the closer two features are located, the easier it may be to "confuse" the two in the presence of errors, e.g., errors produced during the determination of the feature's location or if features do slightly change. There may be singular features with large distance to their neighbors and other features which are densely crowded. In the first case, these may be candidates for 'sheep', i.e., features which are easily recognizable  [Doddington, Sutton]. The distances alone and the number after tightly neighboring features do not allow a classification due to Doddington's zoo. This situation changes, if one adds individual error balls around each feature point which characterize all errors due to non-ideal behavior of the features and their measurement. This will be considered later.

Fig. 3.4: Biometric features as points in a 2D feature space

If the number of dimensions of the feature space is larger than the number of feature components, it is even possible that each feature has the same distance to all of its neighbors. In this case the geometrics can be described by an equilateral K-simplex. This is one case of perfect equality between all features. Using the discrete metric is another method to create equality between all features, irrespective of dimensionality, but possibly accompanied with other drawbacks.

If the dimension is small compared to the number of features and the Euclidean metric is chosen, at least an approximate constant distance to the nearest neighbors should be possible. In this case the number of features in a K-dimensional ball increases with the power of K of the ball radius.

3.2 Enrolment - Verification - Identification

Throughout this paper, we approach the reality by a second way which may meet practice more consequently. Instead of using the virtual index, a user ID is assigned and stored during enrolment. This user ID is nothing but an additional feature component belonging to a subject, however with different usage. For example, user IDs may be public and usually are unique. This way every verification may be performed at least partly as an identification: The subject has to present the user ID as first and the biometric feature as second component. For entering the user ID, a keyboard or a smartcard may be used among other ways. Then the user ID is compared with all stored reference user IDs to see if one matches. If so, in the next step the accompanying biometric part of the reference is compared with the presented biometric feature component to verify that the biometric feature components do match, too. In practice, most verifications are performed this way. The main reason for this fact is that text strings for user IDs are found and checked much faster than the biometric part of a feature. For that reason, a verification is usually much faster than a pure biometric identification where the biometric part of all reference features is to be compared with a candidate until a match is found. [Bromba2007]

In the presence of errors, enrolment may be more sophisticated than simply storing the first available feature sample. Especially stochastical errors may lead to the circumstance that all captured features from the same biometric characteristic are more and less different. As a consequence, the question arises which of the incoming features of subsequent capturings will deliver the best recognition rates. This question is treated later.

The most important property of a (digitally stored) reference feature is its stability over time (unless update strategies are used to improve the reference with each recognition). While stochastical errors in new samples of a characteristic will change with each capturing, the stochastical errors in the reference are stored and thus fixed. For that reason, optimal enrolment strategies try to find the stochastically best reference, typically by processing a series of capturings. As a consequence, enrolment transactions will nearly always take more time than a recognition transaction. For the feature space F of all possible features the consequence is the introduction of the subset F' of reference features. Each reference feature should be as near as possible to the true ("error-free") feature of a certain characteristic. To indicate that a feature is a reference, we sign it or its index by an apostrophe, e.g., fi' or di'j where feature no. i' is the reference belonging to characteristic no. i. Generally, the following relations hold:

fi ∈ F ⊂ F (3.3)

fi' ∈ F' ⊂ F (3.4)

where F is the set of all used features. Only in the ideal case we have F = F'.

4. Biometric Performance Measures

4.1 Definition of Match and Non-Match

F is the space of all features being comparable by a specific recognition system as defined above. d is any distance measure over F with values in D. Subjects si ∈ S shall have the same index i as the corresponding feature fi ∈ F when defining an individual ⓘi, see (2.3). Finally, let F' be the space of all enrolled features and F the space of all features, enrolled subjects are able to assume.

As bilateral match rate MRij we define the probability of a successful recognition when comparing features fi(ω1) and fj(ω2) where ω1 and ω2 are independent outcomes of the sample space Ω [Wikipedia].

MRij ≡  MR(fi,fj) ≡ P{dij ≔ d(fi(ω1), fj(ω2)) ≤ th | fi(ω1), fj(ω2) ∈ F; ω1, ω2 ∈ Ω}  (4.1)

In the absence of any errors ("noise"), all distances are deterministic, i.e., the probabilities are either 1 or 0:

MRij = 1 ⇔ dij ≤ th (4.2)

MRij = 0 ⇔ dij > th (4.3)

Generally, we have symmetry:

MRij = MRji (4.4)

MR11 MR12 MR13 MR14 ... MR1N MR21 MR22 MR23 MR24 ... MR2N MR31 MR32 MR33 MR34 ... MR3N MR41 MR42 MR43 MR44 ... MR4N . . . . ... . MRN1 MRN2 MRN3 MRN4 ... MRNN = 1 0 0 0 ... 0 0 1 0 0 ... 0 0 0 1 0 ... 0 0 0 0 1 ... 0 . . . . ... . 0 0 0 0 ... 1 (4.5)

Until now we have considered the term match rate without interpretation. There are four possibilities to define:

- If two samples of the feature of the same subject match, this is a correct match
- If two features of two different subjects match, this is a false match
- If two samples of the feature of the same subject do not match, this is a false non-match
- If two features of two different subjects do not match, this is a correct non-match

The terms false and correct obviously do not mainly depend on the features but on the subjects, i.e., the subjects belonging to the features compared. Let us assume that for individual ⓘm ≔ (sm,f'm) the reference feature f'm is stored during enrolment. We have 4 possibilities for comparisons (⇆ means that the corresponding subject on the left tries a recognition with the stored subject's feature on the right):

1. (sm,fm) ⇆ (sm,f'm) - fm ∈ F, f'm ∈ F', subject sm tries to get recognized
2. (sm,fn) ⇆ (sm,f'm) - fn ∈ F, f'm ∈ F', subject sm tries to get not recognized (hiding)
3. (sn,fm) ⇆ (sm,f'm) - fm ∈ F, f'm ∈ F', subject sn tries to get recognized (counterfeiting, impersonation)
4. (sn,fn) ⇆ (sm,f'm) - fn ∈ F, f'm ∈ F', subject sn tries to get recognized (using his genuine feature for the "wrong account")

The following definitions only refer to the cases 1 and 4!

Using the previous definitions, if i=j, the bilateral match rate is equivalent to the individual correct match rate CMRii of characteristic no. i. If i≠j, an acceptance would be a failure and thus the probability will be called bilateral false match rate FMRij.

CMRii ≡ MRii (4.6)

FMRij ≡ MRij if i ≠ j (4.7)

As bilateral non-match rate NMRij we define the probability of a failing recognition when comparing features i and j. If i=j, we have a failure and the bilateral non-match rate is equivalent to the individual false non-match rate FNMRii of characteristic i. If i≠j, a non-match is expected and thus will be called bilateral correct non-match rate CNMRij.

If we take the mean value over the individual rates of all N comparisons, the global correct match rate CMR and the global non-match rate FNMR results.

CMR ≡ 1 N N i=1 CMRii (4.8)

FNMR ≡ 1 N N i=1 FNMRii (4.9)

It makes no sense to choose N (in this case the number of features compared) larger than the number of possible features (|F|). On the other hand, N should be much larger than the number of subjects enrolled to the recognition system (|S|) since we are also interested in false matches caused by the features of "external" subjects:

|S| ≪ N ≤ |F| (4.10)

Typically, the size N will be chosen differently for the determination of CMR and FNMR. Since we have defined the match rates as probabilities, noisy features require a practical estimation. This is done by independently repeating a comparison as often as reasonable, say K times, and build the ratio between the number of matches, say M, and the number of trials (K) such that

MRij ~ M/K (4.11)

If we take the mean value with respect to j over the bilateral rates of all N-1 independent comparisons with different index i≠j, the individual false match rate FMRi of characteristic i and the individual correct non-match rate CNMRi of characteristic i results.

FMRi ≡ 1 N−1 N j=1,i≠j FMRij (4.12)

CNMRi ≡ 1 N−1 N j=1,i≠j CNMRij (4.13)

The sums can be simplified, if we set CNMRii = FMRii = 0.

Finally, a global false match rate FMR and a global correct non-match rate CNMR can be defined by taking the mean value over all N individual false match and correct non-match rates.

FMR ≡ 1 N N i=1 FMRi (4.14)

CNMR ≡ 1 N N i=1 CNMRi (4.15)

Generally, we have the relationships

FNMRii = 1 − CMRii FNMR = 1 − CMR (4.16)

FMRij = 1 − CNMRij FMRi = 1 − CNMRi FMR = 1 − CNMR (4.17)

In the case of stochastical behavior of a feature, enrolment may have the effect to create a new element in the feature space. The main property of this element is that it usually is purely deterministic and thus may have completely different stochastical properties. As a result, the match rate definition (4.1) must be modified such that

MRi'j ≡  MR(fi',fj) ≔ P{di'j ≔ d(fi', fj(ω)) ≤ th | ω ∈ Ω } (4.18)

The trick is to pretend that fi' is deterministic since it does not change during normal operation of the biometric recognition system. However, for comparing different systems this may lead to misinterpretations since every new enrolment may deliver new probabilities MRi'j. Thus, when estimating the failure rates, this effect must not be forgotten. A generalization can be given by the following definition:

MRi'j ≡  MR(fi',fj) ≡ P{di'j ≔ d(fi'(ω1), fj(ω2)) ≤ th | ω1∈Ω1⊂Ω2, ω2∈Ω2}  MRij' ≡  MR(fi,fj') ≡ P{dij' ≔ d(fi(ω1), fj'(ω2)) ≤ th | ω1∈Ω1, ω2∈Ω2⊂Ω1} (4.19)

For MRi'j, (4.1) is covered by the situation that Ω1 = Ω2, while (4.18) matches the case whereΩ1 only contains one element of Ω2. Note that the probability MRi'j itself is stochastic if Ω1 is a random subset of Ω2!

To determine the failure rates in ordinary biometric recognition systems, the distance between enrolled reference (i.e., fixed) feature and request feature is measured as in Example 3.1. This yields a different distance matrix with di'j ≔ d(fi',fj), where we mark the reference feature by '. fi' and fj are the reference and actually measured features, respectively.

d1'1 d1'2 d1'3 d1'4 ... d1'N d2'1 d2'2 d2'3 d2'4 ... d2'N d3'1 d3'2 d3'3 d3'4 ... d3'N d4'1 d4'2 d4'3 d4'4 ... d4'N . . . . ... . dN'1 dN'2 dN'3 dN'4 ... dN'N (4.20)

In contrast to matrix (3.1), di'j will generally only be equal to dj'i (i.e., symmetric) if fi' = fi and fj' = fj. Hence, in the stochastical case, the distance matrix will not be symmetric. As a result, the match rate matrix will generally not be symmetric, too:

MR1'1 MR1'2 MR1'3 MR1'4 ... MR1'N MR2'1 MR2'2 MR2'3 MR2'4 ... MR2'N MR3'1 MR3'2 MR3'3 MR3'4 ... MR3'N MR4'1 MR4'2 MR4'3 MR4'4 ... MR4'N . . . . ... . MRN'1 MRN'2 MRN'3 MRN'4 ... MRN'N (4.21)

This is a consequence of the general case which need not be symmetric in this sense:

MRji' = MRi'j ≠ MRj'i = MRij' (4.22)

To keep our individual and global failure rate definitions unique, we always assume the first index to be enrolled (if at all).

In the following examples we show the impact of non-ideal biometric features regarding properties BCR1 - BCR4.

Example 4.1: Weak uniqueness: Twins If the biometric characteristic is completely genotypic [BioFAQ], monozygotic (identical) twins theoretically may show exactly the same features. This could especially be the case in DNA investigations. For the failure rates this may have the following impact. Let SN be the set of N subjects SN ≔ {s1, s2, s3, ..., sN} and si and sj, i ≠ j, being monozygotic twins with the same feature f≔fi = fj and without random errors such that ⓘi = (si, f) and ⓘj = (sj, f). Then the distance dij ≔ d(fi,fj) = d(f,f) is equal to zero although a non-zero result would be required for authentication systems. That is, even with a threshold of th=0, ⓘi and ⓘj are indistinguishable and we get a false match case with a bilateral FMRij = FMRji = 1! All other bilateral false match rates are zero. If N subjects are expected, the individual false match rate of twin ⓘi would obviously be FMRi = 1/(N-1). The same holds true for twin ⓘj. If N approaches large values, FMRi will approach zero. To get the global FMR, the mean value over all N individual false match rates has to be taken. The result is FMR = 2/(N(N-1)) if there are only two twins (i.e., one twin pair) in the ensemble. If there are more twins, the result will differ. If we expect 1 average monozygotic twin pair among 243 people (this seems to be a good estimation to real life [Daugman]), there are 2N/243 twins in the ensemble which all have an individual FMRi of 1/(N-1). This results in a global FMR of (2N/243)/(N(N-1)) = 2/(243(N-1)). That is, for large N, FMR again approaches 0 although not with order N² as in the single-twin pair case. This is in contrast to the minimum FMR of 0.82% stated in [Daugman].

Example 4.2: Weak permanence If the condition of ideal permanence is replaced by time-dependence this means that when capturing a biometric characteristic at different time instants even without measurement failures, more and less slightly different features may result. If so, the biometric recognition system may fail to recognize the characteristic if the threshold is selected too low. On the other hand, other characteristics may also vary. This will have two effects: 1. The minimum  threshold should be larger than the greatest variation of the feature of a certain biometric characteristic and 2. the minimum distance between features of different characteristics will increase. If both begin to overlap, no threshold can be chosen anymore without producing recognition errors. Permanence properties may differ for different characteristics. Fig. 4.1: Stochastical fluctuations of the features in a 2D feature space

Example 4.3: Weak measurability The most apparent impact of non-perfect measurability is the introduction of errors, usually considered as stochastical errors. Such errors introduce stochastical position errors of a feature point in a K-dimensional feature space F, just in the same way as time-depending features will do. Such errors can be described by a K-dimensional cloud which may follow, e.g., a K-dimensional Gaussian distribution. Rather than a point, a feature now should be visualized as a cloud with declining intensity. The intensity then will be a measure for the probability of presence of the feature point at a certain location in F. Such a cloud may overlap with the clouds of other features. This way, no threshold can be found without producing non-zero recognition errors. Fig. 4.2: When averaged over measurements, feature locations may be represented as spatial distributions Furthermore, the distribution will depend on the individual characteristic. This will lead to clouds of different size and shape, as Figure 4.3 indicates. This is the worst case with respect to examination of failure rates since most failure rate definitions or at the least the approximation of their statistical significance make idealizing assumptions which do not hold in reality. Fig. 4.3: Different features may show different distributions Compared to weak permanence, weak measurability does not depend deterministically on time. That is, for weak permanence, two measurements at the same time instant will exactly show the same result while for weak measurability this need not be the case. However, a real biometric characteristic is affected by both shortcomings. Additionally, measurability may change over time. All combined effects can be described by a time-varying multi-dimensional joint probability distribution.

Example 4.4: Human height: Distance errors between two individuals Let us assume two individuals with different heights. The feature space F can be defined as F = ℝ⊕. The distance space D, assuming the Euclidean metric, also equals or is a subset of ℝ⊕. For determining the FNMR, the first characteristic is captured twice. For determining the FMR, the second one is captured once. The measurements of the height features h1 and h2 are error-prone and show a continuous joint probability density fh1,h2 and the continuous singular densities fh1 and fh2, respectively. We assume stochastical independence, such that fh1,h2(x,y)= fh1(x)fh2(y). fh1 fh2 0 ╷ h1 ╷ h2 Fig. 4.4: Heights of two individuals with feature densities and mean values h1, h2 We consider two cases:  1. One sample each of two different characteristics h1 and h2 are compared to determine the bilateral FMR12.  2. Two samples each of the same characteristic, respectively their features h1 and h2, are compared to yield the FNMR.  Both cases assume that enrolment is repeated with each measurement and does not differ from sampling for recognition. In the case of different characteristics, the distance between the two samples is given by the random variable  d12≔ d(h1,h2) ≔ |h1(ω1)−h2(ω2)| = |h2(ω2)−h1(ω1)| = d(h2,h1) (4.23) The bilateral FMR12 is then given by FMR12(th)  ≔ P{d12 ≔ |h1(ω1)−h2(ω2)| ≤ th | ω1, ω2 ∈ Ω } ≔ P{d12 = h1(ω1)−h2(ω2) ≤ th | ω1, ω2 ∈ Ω} − P{d12 = h1(ω1)−h2(ω2) ≤ −th | ω1, ω2 ∈ Ω } = ∫]-∞,th]fh1-h2 − ∫]-∞,-th]fh1-h2 = (∫]-∞,th] − ∫]-∞,-th])(fh1 ∗ f-h2) = ∫]-th,th](fh1 ∗ f-h2) = ∫]-th,th](fh1 ∗ Ⓢfh2) (4.24) where ∗ denotes the convolution of the two distance density functions fh1 and f-h2 and Ⓢ is the mirror operator such that Ⓢf(x) ≔ f(-x) for every f and x. As a consequence, FMR12(0) = 0 and FMR12(∞) = 1. Case 2 is treated in Example 4.5.

Example 4.4 shows us a possibility how to reduce failure rates. Using the same threshold th, FMR12 can be minimized, if the densities fh1 and fh2 of the features h1 and h2 are as narrow as possible and the threshold is smaller than die distance of the true values of the features.

Example 4.5: Human height: Distance errors between two measurements of the same individual In the same way as shown in Example 4.4, the individual Correct Accept Rate CMRi may be determined under the assumption that two subsequently taken samples h1(ω1) and h1(ω2) of the same characteristic are stochastically independent with probability density fh1,h1' (x,y) = fh1(x)fh1'(y) and fh1 = fh1'. This corresponds to the case where the first sample is taken during enrolment and the second one for authentication. However, measurements and processing for enrolment and recognition are exactly equal and are repeated both for each comparison. Then 1 − FNMR11(th) ≕ CMR11(th)  = P{d11 ≔ |h1(ω1) − h1(ω2)| ≤ th | ω1, ω2 ∈ Ω} = ∫]-th,th](fh1 ∗ f-h1) = ∫]-th,th](fh1∗ Ⓢfh1) (4.25) As a result, FNMR11(0) = 1 and FNMR11(∞) = 1. A similar result may be derived for h2, which need not have the same density as h1.

Since usually one of the features stems from enrolment, an optimized enrolment would be able to reduce the FNMR significantly, if the enrolled feature is nearby the "true" feature.

Example 4.6: Human height: Enrolment In Example 4.4 and 4.5, no explicit enrolment was assumed, i.e., two arbitrary stochastically independent samples were directly compared and no special enrolment processing was performed. For each measurement experiment completely new samples were taken. This is not common practice in biometric systems. Most biometric systems use one specific initial enrolment which is retained for every new comparison. To model this procedure we assume that a single sample of h1, say h'1, is taken as reference and use this specific sample for each new comparison. As a consequence, the reference h'1 must be regarded as deterministic. Nevertheless, its stable value is arbitrary and cannot be predetermined before enrolment. As a consequence, its distribution can be approximated by the density distribution fh'1 = δh'1 (4.26) where δh'1 is a delta distribution with peak at location h'1. Recycling the results from Example 4.4, we get FMR1'2(th) = ∫]-th,th](δh'1 ∗ f-h2) (4.27) where h'1 and h2 stem from different individuals. For the special case that the mean value E{-h2} expresses as translation of the density f (example: Gaussian density), we get FMR1'2(th) = ∫]-th,th](δh'1 ∗ f-h2) =∫]-th,th]fh'1-h2 (4.28) If we furthermore assume that h1 and h2 come from the same individual, i.e., h2 = h1, the result would be 1 − FNMR1'1(th) = ∫]-th,th](δh'1 ∗ f-h1) =∫]-th,th]fh'1-h1 (4.29) From (4.29) we can conclude at least for Gaussian densities that FNMR1'1 can be minimized if h'1 equals the expectation of h1 (= E{h1}).

Example 4.7: Human height: Enrolment: Monozygotic twins For monozygotic twins we expect exactly the same height features and feature densities. From (4.24) and (4.25) we then can conclude that the bilateral FMR12(th) is equal to CMR11(th) ≔ 1−FNMR11(th) if all measurements are stochastically independent. In the case of a continuous error density, FMR12(th) will not only assume the discrete values 0 and 1 as for the error-free situation but any value between 0 and 1 depending on threshold. Now let us assume that individual ⓘ1 has been enrolled using a fixed sample h'1. If we take one sample each from individual ⓘ1 and ⓘ2 to determine FNMR1'1(th) and FMR1'2(th), we see from Example 4.6 that the relationship FMR1'2(th) = 1−FNMR1'1(th) remains valid. Although the FMR will change with varying threshold, FNMR will change in the opposite direction. At least for this case, enrolment will have no general influence in the case of monozygotic twins! Generally, if we choose a threshold unequal 0 for the monozygotic twin case, realistic densities yield a FMR smaller than 1. This may lead to the wrong interpretation that under certain circumstances, individuals can even be separated if their true features are equal. From the definition of equal error rate (EER12 = FMR12(the) = FNMR1(the), where the is the unique threshold where equality appears, if possible) we conclude that EER always is ½. This is indeed the worst case EER for recognition, see [BioFAQ].

Example 4.8: Feature space with discrete metrics: Passwords as limit case of a behavioral biometric characteristic Let S be the set of all possible and S the subset of enrolled subjects. F shall be the feature space of all possible passwords of a desired class and F' ⊂ F the subset of all enrolled passwords. The feature space F of passwords is a discrete space which can be equipped here with the discrete metrics δ(hi,hj) ≔ { 0 if hi = hj 1 if hi ≠ hj (4.30) Passwords may be unique or not, depending on enrolment. There are different enrolment methods: 1. Passwords may be selected randomly by the user and then be passed to the service provider 2. Passwords may be assigned to the user by the service provider 3. Passwords may be selected randomly by the user and are then checked for uniqueness by the service provider Obviously, method 1 may create multiple occurrences of the same password such that hi = hj for subjects si ≠ sj, while methods 2 and 3 can simply avoid this situation, provided the number of possible features |F| is equal to or larger than the number of enrolled subjects |S|. In the case of unique passwords, no failures are introduced by enrolment. However, failures may occur during identification. There are two types of failures: The authorized subject (genuine) sn mistypes or confuses the password hn A non-authorized subject (impostor) tries a random or "stolen" password If the genuine mistypes a password, the desired access may fail, producing a false non-match. In a scenario where the password is the only identifier without user ID, the false password at the same time may match a foreign one, leading to a false match with another genuine. That is, a genuine may be an "impostor", too, even during the same transaction. Obviously, a system without user IDs which is only based on unique passwords will be more reliable, if the number of possible subjects is much smaller than the number of possible passwords: |S| ≕ K ≪ |F| (where F is the feature space of all possible passwords with certain properties such as length or character set). In the hypothetic error-free case, the failure rates for enrolment situation 2 (passwords are uniquely assigned to the user by the service provider) would be given by FMRij(th) = 0 if th < 1, FMRij(th) = 1 if th ≥ 1 (4.31) FMRi(th) = 0 if th < 1, FMRi(th) = 1 if th ≥ 1 (4.32) FMR(th) = 0 if th < 1, FMR(th) = 1 if th ≥ 1 (4.33) Since D = {0,1}, the best (and only useful) threshold th here is 0 which is selected from now on. In practice, several FMR failure types are to be considered, e.g.: 2A. One of |S| unique passwords becomes known to one impostor 2B. All impostors select a random password fn ∈ F to match a certain user ⓘm = (sm, fm) where fm ∈ F' In case 2A, we denote the impostor as Ⓘnm ≔  (sn, fm) with n ≠ m. I.e., subject sn ∉ S is "stealing" feature fm which actually belongs to subject sm ∈ S, resp., genuine Ⓖm ≔ ⓘm = (sm, fm).  As a result, the bilateral false match rate FMRij(0) is easily seen to be 1 if (i = m, j = n, i ≠ j) and 0 else. For the individual and global false match rates this means FMRi(0) = 1/(N-1) if i = m and 0 else (4.34) FMR(0) = 1/(N(N-1)) (4.35) where we choose N ≔ |S| = |F'| ≤ |F|. The resulting failure rates only meet the special case 2A and cannot be generalized. This is easily seen when trying to understand the FMR definition in this case. For example, FMRi expects that password fi is tested against all other genuine passwords fj with i ≠ j. Only if i coincides with the index of the stolen password (m) a match happens. In this scenario we obviously expect one impostor with stolen password and all other impostors with their genuine password. This situation can be compared with the example of non-distinguishable twins where only one twin pair is present in the whole ensemble S (Example 4.1). The difference is a factor 2 which comes from the assumption that the impostor has stolen one password while the victim has not done the same with the password of the identity thief. In situation 2B, randomness is introduced by guessing passwords. The match rates are then given by the probability that two passwords coincide. Especially, FMRij(0) = p(fi = fj) (4.36) This probability is equal to zero if either fi or fj ∉ F'. If both passwords belong to F', it depends on the way how the service provider has assigned the passwords and what the impostor knows about it. If we expect a uniform distribution, the probability is simply given by FMRij(0) = p(fi = fj) = |F'|-1 = 1/K if fi and fj ∈ F' FMRij(0) = 0 else (4.37) To determine the individual and global failure rates, we have first to determine the number N of comparisons used for taking the average. If the number of possibilities is finite, N should be set equal to the number of possibilities which is defined by |F|. For example, if F is the space of 4-digit numbers, then N ≔ |F| = 10000. Generally, we get for enrolled individuals FMRi(0) = (1-1/K)/(N-1) if fi ∈ F' and FMRi(0) = 0 else (4.38) If K = N, the well-known result FMRi(0) = 1/N is given. FMR(0) = (K-1)/(N(N-1)) (4.39) If K = N, again FMR(0) = 1/N results. Refinements of this consideration encompass cases where attacking passwords are not selected purely random but with prior knowledge about the probability of being used or not. (We had assumed a uniform  distribution.)

Example 4.9: Simple feature space with stochastical features: Hat Let us assume two individuals ⓘ1 and ⓘ2 with the following properties: ⓘ1 carries a hat with 40% probability and ⓘ2 carries a hat with 30% probability. This may be interpreted as a kind of behavioral biometric characteristic. We try to distinguish these two individuals solely by the feature components "carrying a hat" and "carrying no hat". Furthermore we use the discrete metric and consider all four enrolment states for the reference features f1' and f2'     f1' f2' Reference Distance d(f1',f2') E1 NH NH 0 E2 H NH 1 E3 NH H 1 E4 H H 0 The two individuals share both features with different probabilities which are represented by two points in a one dimensional feature space F which contains two values: "hat" and "no hat".     f1' f2' p1'1(0) p1'2(0) p2'1(0) p2'2(0) p1'1(1) p1'2(1) p2'1(1) p2'2(1) E1 NH NH 0.6 0.7 0.6 0.7 0.4 0.3 0.4 0.3 E2 H NH 0.4 0.3 0.6 0.7 0.6 0.7 0.4 0.3 E3 NH H 0.6 0.7 0.4 0.3 0.4 0.3 0.6 0.7 E4 H H 0.4 0.3 0.4 0.3 0.6 0.7 0.6 0.7   f1' f2' pG(0) pG(1) pI(0) pI(1) FNMR(0) FNMR(1) FMR(0) FMR(1) E1 NH NH 0.65 0.35 0.65 0.35 0.35 0 0.65 1 E2 H NH 0.55 0.45 0.45 0.55 0.45 0 0.45 1 E3 NH H 0.45 0.55 0.55 0.45 0.55 0 0.55 1 E4 H H 0.35 0.65 0.35 0.65 0.65 0 0.35 1 This example also shows the large influence of enrolment. Intuitively, E2 delivers the best enrolment when regarding the failure rates. From the reference distances in cases E1 and E4, no selectivity is to be expected because both subjects are enrolled with the same feature vector. E2 tries to consider the fact that ⓘ1 is more often met with hat than ⓘ2. E3 is something like the converse of E2.

Example 4.10: Simple feature space with stochastical features: Jacket Let us assume two individuals ⓘ1 and ⓘ2 with the following properties: ⓘ1 carries a jacket with 10% probability and ⓘ2 carries a jacket with 80% probability. This may be interpreted as a kind of behavioral biometric characteristic. We try to distinguish these two individuals solely by the feature components "carrying a jacket" and "carrying no jacket". Furthermore we use the discrete metric and consider all four enrolment states for the reference features f1' and f2'     f1' f2' Reference Distance d(f1',f2') E1 NJ NJ 0 E2 J NJ 1 E3 NJ J 1 E4 J J 0 The two individuals share both features with different probabilities which are represented by two points in a one dimensional feature space F which contains two values: "jacket" and "no jacket".     f1' f2' p1'1(0) p1'2(0) p2'1(0) p2'2(0) p1'1(1) p1'2(1) p2'1(1) p2'2(1) E1 NJ NJ 0.9 0.2 0.9 0.2 0.1 0.8 0.1 0.8 E2 J NJ 0.1 0.8 0.9 0.2 0.9 0.2 0.1 0.8 E3 NJ J 0.9 0.2 0.1 0.8 0.1 0.8 0.9 0.2 E4 J J 0.1 0.8 0.1 0.8 0.9 0.2 0.9 0.2   f1' f2' pG(0) pG(1) pI(0) pI(1) FNMR(0) FNMR(1) FMR(0) FMR(1) E1 NJ NJ 0.55 0.45 0.55 0.45 0.45 0 0.55 1 E2 J NJ 0.15 0.85 0.85 0.15 0.85 0 0.85 1 E3 NJ J 0.85 0.15 0.15 0.85 0.15 0 0.15 1 E4 J J 0.45 0.55 0.45 0.55 0.55 0 0.45 1 This example also shows the large influence of enrolment. Intuitively, E3 delivers the best enrolment when regarding the failure rates. From the reference distances in cases E1 and E4, no selectivity is to be expected because both subjects are enrolled with the same feature vector. E3 tries to consider the fact that ⓘ1 is less often met with a jacket than ⓘ2. E3 is something like the converse of E2.

4.2 Separability

Since multidimensional probability density functions are more difficult to handle, we focus on the one-dimensional densities of the distance between two features. The probability density function of the distances depend on the character of the noise, the distance measure, the number of features, and the feature dimension.

We start with the case of finite many different distances in F. For example, when using the discrete metric, only the values 0 and 1 can appear. For each of these values their frequency can be determined either by knowledge of the stochastical properties or by measurements.

Let d: F x F ↦ DN ⊂ ℝ⊕ be a distance measure and DN be a discrete space with N different positive real numbers. Then the bilateral distance density functions are defined by

pi'j(x) ≡  P{ω1, ω2 ∈ Ω | di'j ≔ d(fi'(ω1), fj(ω2)) = x ∈ DN} (4.40)

If i'=j, pi'j defines the bilateral distance density function for the feature outcomes of the same characteristic. The case i'≠j represents the probabilities between the two features i' (reference) and j (request) of disjoint characteristics. From the bilateral densities the global genuine and global impostor distance densities can be derived:

pI(x) ≡  1 N(N−1) N i=1   N j=1 j≠i pi'j(x) (4.41)

pG(x) ≡  1 N N i=1 pi'i(x) (4.42)

Generally, the failure rates can be calculated from these densities. For the global FNMR and FMR we get

FNMR(th)  = 1 −     x∈DN x≤th pG(x) with   x∈DN pG(x) = 1 (4.43)

and

FMR(th)  =     x∈DN x≤th pI(x) with   x∈DN pI(x) = 1 (4.44)

The bilateral and individual rates can be defined correspondingly. If D is a continuous set D = [0,K] with K ∈ ℝ⊕, such that pG, pI, FNMR, FMR are functions D → [0,1], the sums have to be replaced by integrals:

FNMR(th) D= 1 − ∫[0,th] pG with D∫DpG = 1 and th ∈ D (4.45)

and

FMR(th) D= ∫[0,th] pI with D∫DpI = 1 and th ∈ D (4.46)

The two FNMR(th) and FMR(th) curves can be combined into one curve when using the threshold variable th as common implicit parameter. A plot of FNMR against FMR is usually called Detection Error Tradeoff (DET) curve. If CMR is plotted against FMR, the diagram is usually called Receiver Operating Characteristic [Wikipedia]. This term has originally been created in radar signal detection theory. The advantage of the DET is that it shows a global view of the performance of a recognition system without scaling effects of the threshold variable.