Fingerprint ridge distance computation methodologies

Pattern Recognition 33 (2000) 69}80

Fingerprint ridge distance computation methodologies Zs.M. KovaH cs-Vajna*, R. Rovatti, M. Frazzoni DEIS, University of Bologna, viale Risorgimento 2, 40134 Bologna, Italy Received 16 October 1997; received in revised form 25 January 1999; accepted 25 January 1999

Abstract The average ridge distance of "ngerprint images is used in many problems and applications. It is used in "ngerprint "lter design or in identi"cation and classi"cation procedures. This paper addresses the problem of local average ridge distance computation. This computation is based on a two-step procedure: "rst, the average distance is de"ned in each signi"cant portion of the image and then this information is propagated onto the remaining regions to complete the computation. Two methods are considered in the "rst step: geometric and spectral. In the geometric approach the central points of ridges are estimated on a regular grid and straight lines passing through these points and parallel to the ridge directions are used. The second method is based on the computation of harmonic coe$cients leading to e!ective estimates of the average ridge period. In order to complete the average distance map a di!usion equation is used so that maps with minimum variations are favored. Finally, some experimental results on NIST SDB4 are reported.  1999 Patern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. Keywords: Image pre-processing; Fingerprint; Ridge distances; Spectral analysis; NIST SDB 4

1. Introduction Each person has a unique set of "ngertips. Thus, "ngerprints have been used as a personal signature since ancient times [1,2]. The scienti"c foundations of "ngerprint employment for personal identi"cation were laid by F. Galton (1822}1916), H. Faulds (1843}1930) H. Wilder (1864}1928) and H. Poll (1877}1939). Galton [3] pointed out the uniqueness and permanence over time of "ngerprint geometry. Nowadays "ngerprints are mainly used in three areas, namely forensic science [4,5], security clearance ("nancial transactions or access to restricted areas), and anthropological and medical studies (speci"c diseases and genetic features) [6]. The police traditionally collect "ngerprint impressions of suspects' inked "ngers on paper, hence introducing

* Corresponding address. DEA - Facolta' di Ingegneria, University of Brescia, via Branze 38, 25123 Brescia, Italy. Tel.: #39-030-3715437; fax: #39-030-380014. E-mail addresses: [email protected] (Z.M. KovaH csVajna), [email protected] (R. Rovatti)

noise because of insu$cient or excess ink, and irregularities on the paper surface. Noise is also present in latent "ngerprints, no matter how they are detected and recorded. Furthermore, latent "ngerprints are often distorted by non-uniform pressure or movement. Both noise and imperfect recording techniques make the matching task very di$cult. Security clearance systems use sensors to perceive and record a "ngerprint. Though movement-induced distortion is certainly reduced by the cooperation of the person to be identi"ed, noise will still appear as the "ngerprint is recorded in real conditions, when the "nger can be dirty and/or wet. Moreover, irregular translation and rotation generally a!ect various samples of the same "ngerprint because of varied "nger position, "nger moisture and pressure on the recording surface. Fingerprint matching cannot therefore be performed by superimposing two di!erent "nger print images (FPIs). Instead, "ngerprints are matched by comparing ridge patterns, the relative positions of ridge discontinuities (called minutiv) such as ridge endings and bifurcations and the ridge count between minutiv. Other minutiv can be considered as well as pore information or more complex topological structures [5].

0031-3203/99/$20.00  1999 Patern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 1 - 3 2 0 3 ( 9 9 ) 0 0 0 4 0 - 0

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This paper addresses the problem of local average ridge distance computation in FPIs. Though ridge distance is an essential parameter in many procedures in automatic FPI processing and matching systems, so far its computation seems not to have been investigated in detail. In most automatic FPI processing systems, the images are "rst "ltered (regardless of their gray scale or binary nature) to facilitate matching. An important parameter during this step is the "lter window, often a function of `ridge pattern perioda. Formally speaking, ridge period is not the same as ridge distance despite the frequent interchangeability of the two terms in the literature. In fact, the distance from a given ridge to the subsequent one is locally de"ned as the length of the segment connecting the centers of the two ridges along the line perpendicular to the "rst one, while the ridge period is commonly de"ned as the sum of the widths of the ridge and the subsequent valley. Though the di!erence between ridge distance and ridge period can be locally relevant, their averages tend to coincide in patterns with limited curvature. To see how this happens assume that n ridges are present with di!erent widths w ,2, w and separated by N!1 val L leys with di!erent widths v ,2, v . Referring to Fig. 1  L\ it is easy to see that the average ridge distance will be






1 L\ w w G#v # G> G n!1 2 2 G while the average ridge period is 1 L\ (w #v ), G G n!1 G so that their di!erence is ()(w !w )/(n!1). Note that   L this a di!erence vanishes for uniform patterns (with w "w "2"w ) and decreases as more ridges are   L considered, i.e. for increasing n. Many contributions mention and exploit the ridge distance estimation and stress its weight in FPI process-

Fig. 1. Comparison between ridge distance and ridge period.

ing, but, to the best of the present authors' knowledge, none of them addresses its computation in depth. O'Gorman and Nickerson [7] propose a "lter design process specially tailored to gray}scale "ngerprints. Their "lter takes advantage of the assumptions about the underlying pattern usually made by humans when inspecting FPIs. Some "ngerprint image parameters are required so that the "lter is better matched and produces the required image enhancement. One of the key parameters is the ridge pattern period. O'Gorman and Nickerson assume it to be constant over the entire image. It is considered to be either the sum of the maximum ridge width and the minimum valley width or the sum of the minimum ridge width and the maximum valley width. Actually, the authors assume these two quantities to be the same and express ridge period by means of four parameters in order to provide for internal variations of ridge and valley width. In order to improve the performance of their "ltering procedure, O'Gorman and Nickerson propose using the statistical average for ridge distance, depending on the subject class (the smaller the class the better the "lter). To do so, they preselect FPIs, partition them into di!erent classes (men, women and children) and design a di!erent "lter for each class. This approach has two weaknesses: the assumption that the ridge period is constant over the entire image, and the grouping into classes of di!erent people. Both these weakness led to problems which the authors claimed were dependent on incorrect feature detection. Lin and Dubes [6] have developed a system for automatic ridge counting and assume ridge period to be constant. Nevertheless, in the discussion of their results they observe that ridge period variation sometimes jeopardizes the possibility of correct ridge counting. Hung [8] computes ridge period independently for each "ngerprint. He attempts to overcome internal ridge width variation by equalizing it throughout the whole FPI. First he applies a binary "lter that uses an estimate of ridge pattern period. Then a direction map is computed, and from any ridge point the next ridge is searched for by moving perpendicularly to local ridge direction. The ridge pattern period is computed by summing the averages of ridge width and valley width over the whole image. This method su!ers from several problems. First, the high internal variation of ridge and valley width can preclude proper processing in a considerable portion of the image if average values over the whole image are computed. Second, neighboring ridges are usually oriented in di!erent directions so that the exact computation of their spacing would require extensive inspection of the image instead of a local orthogonal search which may give incorrect results. Finally, when noise is present in the form of spurs, holes and short ridges, grossly mistaken ridge and valley widths may be included in the "nal statistics, a!ecting their reliability.

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After "ltering, binarizing and ridge thinning, a ridge skeleton is obtained by all the commonly adopted FPI processing systems. With skeletonized images, ridge distance becomes an even more important feature. In fact, the whole skeleton enhancement process consists of selecting a few dozen minutv with a high con"dence rating and using them for actual "ngerprint matching. Hung himself, as well as Xiao and Rafaat, employ ridge distance in analyzing skeleton structure, in order to remove false minutiae or noisy skeleton patterns. For example, most heuristics for minutv extraction are based on local analysis (close bifurcations, facing endpoints, short ridges) in a neighborhood whose size is a direct function of the ridge distance. From the above discussion, some key factors seem to a!ect the average ridge distance estimation. 1. Variability of ridge distance from one person to another. 2. Variability of ridge distance up to 150 within the same FPI. 3. Variability of ridge direction. 4. High levels of noise that may distort statistics. As an additional feature, any candidate methodology for ridge distance extraction should be applicable to grayscale FPI as it is one of the earliest processing phases. This paper is organized as follows. The following section introduces some basic notations which are common to the methods we propose and discuss. In Sections 3 and 4, two di!erent approaches to average ridge distance estimation are described. They are, respectively, inspired by geometric considerations and by the physical signi"cance of image spectral features. Since all the previously described methods are able to e!ectively estimate the average ridge distance only in some sub-blocks of the original image, a mechanism is needed to extend the estimation to sub-blocks corrupted by noise or with a high curvature pattern. Section 5 deals with this problem and proposes a feasible, formally sound approach to this task. In Section 6 the three methods are compared by analyzing their response when some FPIs from a standard database are processed. Finally, some conclusions are drawn and future directions of investigation discussed.

2. Common notations and assumptions All the procedures analyzed in the following consider gray-level FPIs partitioned into square blocks of N;N pixels. Fig. 2 shows a 256-level 512;512 FPI from the NIST-SDB4 database [9] partitioned into 64 square blocks with N"64. Average ridge distance estimation will be carried out separately in each sub-block to cope with the high variability over the whole FPI. The procedures represent each

Fig. 2. Partition of f0025}6 from the NIST-SDB4 database into zones for average ridge distance estimation.

block with a matrix g , assigning a gray level to each VW pair of integer coordinates x, y3+0,2, N!1,. For each block the two numbers g "min g and

 VW VW g "max g identify the actual range of gray vari  VW VW ation.

3. A geometric approach As ridge distance is a geometric feature of the FPI the most natural way of tackling its estimation relies on recognizing and processing basic geometric entities. The basic entities we are interested in are the central points of ridges and the straight lines passing through them parallel to the local ridge. In fact, wherever in the sub-block these entities can be recognized, analytical computation of the ridge distance is a matter of trivial trigonometric considerations. Hence, we are "rst concerned with recognizing these entities in a FPI which is noisy and irregular. Central ridge points are always located at minima of the gray-scale matrix g , though not every minimum VW corresponds to a ridge center. Even for a relatively small block, an exhaustive search for a gray level minima and the subsequent processing needed to associate subsets of the minima to the ridges they lay on are not advisable from the computational point of view. A regular grid is therefore superimposed on the block as shown in Fig. 3 and local minima are searched for along each horizontal and vertical line.

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Fig. 3. A minima-searching grid for the central block in Fig. 2.

Due to noise and pattern irregularity, not every local minimum is a ridge center; candidate minima must therefore be tested against suitable criteria to ensure that they are true background pixels. To do so a threshold gth is de"ned and only minima at points (x, y) such that g )g are taken as ridge centers. VW  As this is the starting point of the whole procedure, the de"nition of g is critical and deserves careful analysis. In  good quality images, foreground and background are

well distinguished so that many pixels are either bright, i.e. high g , or dark, i.e. low g . If statistical inVW VW formation is collected about the relative frequency of each gray level in such an image, a typical bi-modal histogram is obtained. Many methods for threshold extraction have been proposed in literature (e.g. Refs. [11}13]). As we must perform threshold selection for every row and every column in the grid we cope with the need for a fast procedure devising an histogram-based

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Fig. 4. The bi-modal gray-level histogram of a good image.

Fig. 5. First and second trial for the gray-level threshold g . 

technique combining many aspects of the methods found in literature. Fig. 4 shows a typical gray-level occurrence histogram. It has been obtained by grouping 256 original gray levels into 64 groups so that the ith histogram entry actually accounts for the number of occurrences of gray levels 4i, 4i#1, 4i#2, and 4i#3. In such a case, the natural value for a gray-level threshold discriminating between foreground and background lies somewhere in the valley between the two relevant maxima. Its numerical evaluation exploits a two-step procedure aimed at avoiding statistically negligible contribution to the gray-occurrence histogram. First, the gray-level histogram is collected by analyzing the pixels along the chosen sub-grid line and assigning the *g gray levels between a and a (with a "i*g) to G G> G the ith entry. Then the histogram is smoothed by averaging every entry with its neighbors. Finally, if H is the relative occurrence of the gray-levels grouped G in the ith entry of the smoothed histogram and H "

 max H is their maximum, we de"ne the two indices G G i "min+i : H *H /2, and i "max+i : H *H /2,.  G

  G

 These two indices delimit the statistically signi"cant portion of the gray-level histogram and a "rst threshold g is set to  i #i #1  . g"*g   2 The construction of i and i from the half-mode   horizontal line is sketched in Fig. 5. Note how, even with this well behaved histogram, this "rst estimate may fail to "nd the most natural threshold value, though it certainly identi"es the most promising region. A "nal tuning is therefore needed. To this end we search for local minima of H in a neighborhood of G (i #i #1)/2. In practice, we consider those histo  gram entries i such that H )min+H , H , and G G\ G> "i!(i #i #1)/2")20/*g.  

If such entries exist, the one with the minimum H de"nes a second gray-level threshold g"*g(i#), G   while if no local minima are present in the neighborhood of (i #i #1)/2g is adopted.    Though devised under the hypothesis of a bi-modal gray-level histogram, this method can also be applied to unimodal or multi-modal pro"les, often resulting in a reasonable estimate of the natural gray-level threshold distinguishing foreground pixels from background pixels. Gray-level threshold extraction is performed for each horizontal and vertical line of the grid crossing the subblock so that a set of ridge centers can be constructed consisting of all local minima falling below those thresholds. Once the ridge centers are collected, computation of the local distance requires knowledge of the ridge direction. The method we adopt is similar to the one used in Ref. [8], in which only a discrete number of directions is considered. Fig. 6 illustrates the procedure for a number of distinguishable directions n equal to 8, while practical  implementation employs a resolution of n "16.  Let P be the center of the ridge just discovered. A disk with radius R"10 pixels is considered, partitioned into n equivalent sectors S ,2, S . Then, for each sector   L S we de"ne the two quantities g G"min G g and G

 1 VW [g !g G]

 , p " VWZ1G VW G CS G where the C operator gives the number of pixels in S . G The index p accounts for the presence of foreground gray G levels in S with a quadratic emphasis. Hence, the darker G and more compact the region, the smaller the index, and the ridge direction is taken to be the one associated with the minimum dispersion p , i.e. the line bisecting that G sector. This method can yield a wrong direction when ridges are very close and the darkest pixel of the sector does not

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Fig. 6. Extraction of ridge direction.

Fig. 7. Next ridge search and distance evaluation.

belong to the ridge under examination but to one of its neighbors. Nevertheless, experience has shown that these cases are statistically insigni"cant. Once a central point P and a ridge direction have been extracted we search for the two neighboring ridges forwards and backwards along the line normal to the ridge direction. To do this, the center searching algorithm explained above is employed, accepting only those centers whose distance from the initial ridge center is not greater than a threshold d .

 We are now out to evaluate the distance between each pair of ridges, accounting for the fact that the two are, in general, not parallel. Two lines starting from P are therefore considered, one normal to the original ridge, the other normal to the second ridge. If P is the center of the second ridge on the "rst of the two lines, then the second one marks a point P on the line parallel to the second ridge and passing through P. The distance between the two ridges is de"ned as

Acceptable distances are extracted by repeating the above procedure for each ridge center found along every horizontal and vertical line of the grid. Each element in this distance set is then rounded to the nearest integer. With this integer distance set we may now construct a distance histogram whose ith entry Di accounts for the occurrences of a distance whose rounded value is i. The histogram describes the distribution of distances over the sub-block and the "nal average ridge distance is taken to be the average of the signi"cant portion of this distance distribution. The signi"cant portion is, as before, obtained by de"ning the two indices i and i used to estimate the color   threshold on the gray-level histogram. Once i and  i are de"ned we set the average distance to  G iD dM " GG G G D GG G whenever the number of distances in the signi"cant portion of the histogram ( G D ) is not less than a threGG G shold D JN. If, on the contrary, the number of

 signi"cant distances is not a signi"cant fraction of the sub-block size, the estimate is rejected.

PP#PP d" . 2 Fig. 7 shows a sample application of this technique. The resulting distance d is accepted only if it is greater than threshold d and if the extracted directions of the

 two neighboring ridges are not too di!erent. This prevents false ridges due to noisy pixels in valleys altering the overall distance statistics.

4. The spectral approach Ridge period computation can also be regarded as a spectral analysis problem. In fact, regular two-dimensional patterns may be thought of as a linear combination of

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simple orthogonal periodic signals (called harmonics). The closer the pattern to a given periodic behavior, the greater the coe$cient of the harmonic with that period. As spectral analysis enables the computation of these harmonic coe$cients, it may also give e!ective estimates of the average ridge period. 4.1. Discrete Fourier transform and harmonic coezcients If g is the gray-scale value of the pixel with coordiVW nates x, y3+02, N!1, in a N;N image, the Discrete Fourier Transform (DFT) of g is de"ned as [[10], VW Chap. 8] 1 ,\ ,\ G " g e\pH,6VWST7, (1) ST N VW V W where j is the imaginary unit, u, v3+0,2, N!1, and 1(x,y)(u,v)2"xu#yv is the vector dot product. G is, ST in general, complex. However, if we express G in ST Eq. (1) as G ""G "eH %ST, it obviously follows ST ST that [[10], Chap. 8] 1 ,\ ,\ g " G epH,6VWST7 VW N ST S T ,\ ,\ 2p1(x,y)(u,v)2 " "G "cos #arg(G ) ST ST N S T in which we exploit the fact that, g being real, the VW imaginary part of the right-hand side can be discarded. In the light of this, the image can be understood as a sum of harmonics h (x,y)"cos[2p/N1(x,y)(u,v)2] ST whose phase and amplitude are modulated by the complex harmonic coe$cients G . The set of all the harST monic coe$cients of an image is commonly referred to as its spectrum. In Fig. 8 the three harmonics h (x, y), h (x, y), and   h (x, y) are shown for a 64;64-pixel image. Note how  they reproduce an extremely regular pattern whose peri-





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od can be computed analytically from the expression for h (x,y), resulting in ST d "N/(u#v. ST

(2)

In fact, h (x, y) is maximum for all points (x,y) beST longing to the straight line xu#yv!kN"0 for some integer k. However, the distance between two such straight lines for k and k#1 can be easily computed to give Eq. (2). Several well-known features of the DFT are of great importance in practical computation and deserve to be mentioned. In fact, it can be shown that the two-dimensional set of harmonics and harmonic coe$cients is symmetric and periodic along each axis since we have, for example, G "G and h "h as S>,T ST S>,T ST well as G "G and h "h . A more ,\ST \ST ,\ST \ST global symmetry also exists as G "G and \S\T ST h "h . \S\T ST Finally, two more properties are of interest for our analysis. Let us "rst analyze what happens to the harmonic coe$cients when the original image g is transVW formed into a di!erent image g de"ned as VW g W X W X "g , VW  V> V , W> W , where x, y"0,2, N!1 and the translated image is `wrapped arounda de"ning W aX to be a if 0)a(N, , a!N if a*N and N!a if a(0. Starting from the de"nition of the DFT and of g and with a suitable VW sum splitting it follows that 1 ,\ ,\ G " g e\pH,6 WV> VX , WW> WX ,ST7 VW ST N V W

Fig. 8. Images of the three harmonics h

1 ,\ ,\ " g e\pH,6VWST7e\pH,6 V WST7 VW N V W "G

e\pH,6 V WST7, ST

(x, y), h (x, y), and h (x, y).   

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so that "G """G ". Let A now be an orthogonal 2;2 ST ST matrix and let us consider a modi"ed image

The method we are proposing detects peaks of Q(r) and estimates the average ridge distance in consequence:

g W X "g , VW , VW

1. Compute G from g . ST VW 2. Compute Q(r) for 0)r)N!1. 3. Find r such that Q(r)*Q(r) for any 0(r )

 r)r (N!1, rOr ("nd the position of the lar  gest peak). 4. If r is not a local maximum of Q(r) for 0)r)N!1 estimation is impossible. 5. Find r such that Q(r)*Q(r) for any 0(r )

 r)r (N!1, rOr, rOr ("nd the position of

 the second largest peak). 6. Estimate dM "N/r with con"dence.

where the wrap-around is applied separately to each coordinate of the vector A(x,y). If A(x,y) has all integer coordinates we have 1 ,\ ,\ G " g e\pH,6 WEVWX , ST ST N VW V W 1 ,\ ,\ " g e\pH,6VWRST7. VW N V W From this and from the fact that A\"AR we can easily prove that G "G , i.e. that passing from g to g the ST ST spectrum undergoes the same orthogonal transformation. Though the integer assumption on A(x,y) is crucial in the formal development, this result can be extended to image rotations which often feature the transcendental matrix




A"



cosa

!sina

sina

cosa

if a certain degree of approximation is acceptable. 4.2. Ridge distance estimation from the harmonic coezcients The whole procedure relies on a radial distribution function Q(r) de"ned for every integer 0)r)N. To de"ne such a function say that C is the set of coordinates P u, v such that (u#vKr where the K symbol is resolved by rounding to the nearest integer. If CC is the P number of elements of C then Q(r) is de"ned as P 1 Q(r)" CC

"G ", ST P STZ!P

where 0/0 is de"ned to be 0. Recalling the signi"cance of "G " and the expression for the ridge distance resulting ST from a single harmonic (2), it can be intuitively accepted that Q(r) gives the average contribution of the harmonics with ridge distance N/r to the construction of the overall image. Thus, peaks in Q(r) identify the main harmonic contributions and the corresponding ridge distance is likely to dominate in the image pattern. Note how, owing to the two last properties of the DFT, the de"nition of Q(r) is well posed, as it is invariant with respect to translation and rotation of the image. In fact, computation of Q(r) only involves the modulus of the harmonic coe$cients, and the set of coe$cients taken into consideration for each r is obviously invariant through any rotation of the spectrum, i.e. through any rotation of the image.

a min+Q(r)!Q(r), Q(r)!Q(r!1), Q(r) Q(r)!Q(r#1),. From the con"dence expression it can be easily understood that an estimate is considered to be reliable when r is a well-de"ned peak of the radial density function and the second maximum is su$ciently smaller. Rounding is "nally applied to dM to obtain an integer number of pixels. The direction of rounding is biased by the position of the second maximum of Q(r). The three parameters the whole procedure depends on are easily determined from the available data. In particular, once it is known that a typical human "ngerprint has a ridge distance within the interval [d , d ] one may

  set r "N/d and r "N/d . These two para 





 meters set a "ltering mask on the radial distribution function Q(r) that helps to cut out the high-frequency noise contribution (r'r ) and average gray-level ef  fects (r(r ). For typical applications, relying on

 a 500DPI image of the "ngerprint, two reasonable values turn out to be r "3 and r "20.



 The con"dence normalization coe$cient a is only needed when estimates obtained with the spectral method have to be compared with those obtained with other methods. The value of a may be obtained by estimating the ridge distance in ideal images with parallel equidistant gray peaks (see Fig. 8). Then, setting a" Q(r) , max+Q(r)!Q(r), Q(r)!Q(r!1), Q(r)!Q(r#1), where the max is taken over all the considered ideal images, provides a reasonable con"dence normalization coe$cient. This procedure leads to a normalization coef"cient a"2.43. Once that normalized con"dence had been computed we decided to accept only estimations whose con"dence was not less that 0.4.

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5. Ridge map completion All the previously discussed approaches provide local estimation of the average ridge distance. They can report when their estimate is not reliable, because the corresponding image block is a!ected by noise or has too complex a ridge geometry. As a result, an incomplete map of average ridge distances is obtained, in which some zones are assigned an average distance while some are not. Regrettably, ridge distance must be de"ned in each zone to allow "ltering and skeletonization procedures to be applied. To address this problem, let us "rst consider an ideal, continuous image in which a number d(x, y) is associated with a square zone centered on (x, y), which represents the average ridge distance in that zone. Our problem is equivalent to the reconstruction of the function d(x, y) for every allowed x and y, given its values d at certain points G (x , y ). Such a task is well-known to be ill-posed in the G G sense of Hadamard [14] since its solution is neither unique nor depends continuously on the given data. In order to solve it, some constraints re#ecting our knowledge about the actual d(x, y) have to be introduced [15,16], the most natural and commonly adopted being some kind of smoothness enforcement. If we choose to measure smoothness with the secondorder Lebesgue norm of some of the derivatives of d, we may recast the reconstruction problem into the minimization problem min



¹[d](x,y) dx dy " s.t. d(x , y )"d , G G G where ¹[ ) ] is a Tikhonov regularizer, i.e. a positive quadratic di!erential operator implicitly de"ning the concept of smoothness. To complete the map of average ridge distances we choose the simplest Tikhonov regularizer, i.e. the modulus of the gradient operator so that maps which minimize variations are favored. This is quite a natural assumption as ridge distance is known to change little between adjacent zones and these changes are even further smoothed by the local averaging procedure. Thus, we set ¹[d]"(Rd/Rx)#(Rd/Ry) and apply elementary calculus of variations [17] to obtain that maximally smooth d satis"es (Rd/Rx)#(Rd/Ry)"0. Thus, using the Laplacian operator , our problem is to "nd a function d(x,y) such that



d"0,

(3) d(x , y )"d . G G G We may now go back to our discrete problem and solve Eq. (3) for the central points of the blocks for which no reliable estimate was provided. To do so, we "rst translate Eq. (3) into a di!usion problem which is suitable for

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an iterative and discrete solution. That is, we consider d to be the steady-state solution of Rd " d Rt

(4)

which can be discretized both in time and space to obtain dR> R(x, y)!dR(x, y) *t dR(x#1, y)!2dR(x, y)#dR(x!1, y) " (*x) dR(x, y#1)!2dR(x, y)#dR(x, y!1) # . (*y) As *x"*y"*, the iteration of the above formula is stable only if *t/*)1/4 [[18], Chap. 17] and assuming *t"*/4 we obtain dR>DR(x, y)"[dR(x#1, y)  #dR(x!1, y)#dR(x, y#1)#dR(x, y!1)] (5) of which iteration yields the steady-state solution of Eq. (4) as tPR, i.e. the solution of Eq. (3). To enhance convergence when few given data are present we enlarge the number of values of d entering the right-hand part of the discretized equation. To do so it must be kept in mind that the Laplacian operator is invariant under rotation, and follow the same discretization path along the diagonals of the average ridge distance maps dR> R(x, y)"[dR(x#1, y#1)  #dR(x!1, y!1)#dR(x!1, y#1) #dR(x#1, y!1)],

(6)

where it has been noted that moving along diagonals implies a space discretization step (2 times the previous one and thus a double time step. With this, and assuming a piecewise-linear time evolution is assumed for the unknown values of d(x, y), we may average Eqs. (5) and (6) so to take all the eight neighboring zones into account at each iteration.

6. Experimental results The two methods presented in the previous sections have been tested on many di!erent "ngerprint images from a standard database. In particular, we applied them to NIST-SDB4, a set of 2000 pairs of 500DPI FPIs, coded in 8 bit gray scale [9]. As an example let us consider two images from this database labeled, respectively, f0025}06 (see Fig. 2) and s0011}2 (see Fig. 9). As ridge distance is subject to wide variation over the whole "ngerprint, the enhancement procedure is applied block by block. Block sizes from N"50 to 150 were tried

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Fig. 9. The 500DPI "nger print image s0011}2.

Fig. 10. Results of average ridge distance estimation for f0025}6 with the geometric and spectral approaches, local estimation and distance map completion.

and a reasonable trade-o! between the number of blocks in which the procedures produce a reliable estimate (which increases with N) and their ability to track local ridge-distance variation (which decreases as N increases) was found for N in the range [60,100]. For this reason

Fig. 11. Results of average ridge distance estimation for s0011}2 with the geometric and spectral approaches, local estimation and distance map completion.

N was set to 64 so that each 512;512 image was divided into 64 sub-blocks. Ridge distance estimation by means of the geometric and spectral approach is performed for each sub-block. Figs. 10 and 11 report the results of average distance estimation for f0025}6 and s0011}2 with both the Geometric and Spectral approaches. Null entries on the left-hand side tables represent the fact that both the geometric and spectral approaches may fail to give a reliable estimate of average ridge distance. Right-hand side Figs. 10 and 11 report the completed distance maps obeying the smoothness criterion discussed in Section 5 for the two FPI under consideration. A failure in estimation may occur when less than D signi"cant distances can be averaged to obtain the

 "nal estimate (i.e. 0.01;64K41 in the considered cases) or when the normalized con"dence for the spectral estimate is less than 0.4. Though the latter case has a quite straightforward interpretation in terms of noisy or highly irregular patterns, the number of signi"cant distances extracted with the geometric approach is a!ected by many factors. Table 1 shows the incidence of the various decisions on the "nal number of distances. In principle, every local minimum along the horizontal and vertical grid lines in the sub-blocks may lead to two local distances. Yet, only minima under the gray-level threshold are accepted and ridge direction is estimated only for those points not too close to the block border. Moreover, even if the direction

Z.M. Kova& cs-Vajna et al. / Pattern Recognition 33 (2000) 69}80 Table 1 Incidence of di!erent phenomena in the reduction of the number of collected distances Heuristics

Discarded over Discarded over total potential total discarded

Minimum above threshold Block border hit Neighbor search failure Di!erent directions Distance below threshold

22.93 0.77 12.21 7.25 12.17

Total

55.33

41.43 1.40 22.06 13.11 21.99

can be extracted, the search for neighboring ridges may partially or totally fail. For the local distance to be computed, we further require that the direction of the neighboring ridges is not too di!erent from the direction of the "rst one, and local distances themselves are accepted only if they do not fall below a certain d (six pixels

 in the cases under consideration). Table 1 shows the average incidence of these phenomena in the reduction of the number of extracted distances. It can be easily seen how the gray-threshold computation is critical in establishing the `qualitya of the extracted distances, followed by geometrical considerations such as border e!ect avoidance and constraint enforcement on the extracted distance. A substantial agreement with ridge distances extracted by visual inspection has been veri"ed on a random sample of 20 images from the database. Beyond this, the assessment of the performance of the two proposed methods has to consider the whole "nger print recognition system in which the procedures are embedded. To do so we consider a minuti~ extraction system which provides minutiv to a classixer [19]. If the ridge distance is assumed constant and equal to the average over the database, the number of minutiv that are automatically extracted amounts to K75% of the number of real minutiv. When "ltering and skeletonization exploit the estimation of the ridge distance provided by the proposed methodologies this extraction rate is increased to K90%. At an even higher level, we provided the classi"er with one FPI and asked it to "nd the other FPI belonging to the same pair in the database (the `veri"cationa task speci"c of allowance systems). When a "xed ridge distance is assumed and a false-positive rate of not more than 0.05% is required the number of matches is not more than 77% of the total number of FPI. If ridge distance estimation is introduced this ratio is increased to 85% with the same false-positive ratio.

79

7. Conclusion Since the average ridge distance in "ngerprint images is an important parameter in various problems and applications, several methodologies which have proved capable of tackling the problem of its computation were illustrated. The average ridge distance is used in "ngerprint "lter design, and in identi"cation or classi"cation systems. The average distance computation presented in this work is based on a two step procedure: "rst, the average distance is de"ned in each signi"cant portion of the image and then the complete distance map is obtained by propagating the distance values by means of a di!usion equation. We have developed two methods capable of extracting the distance values in the main portions of the image: a geometric method and a spectral one. In the geometric approach central points of ridges are estimated on a regular grid and tangents to the ridges at these points are used. The second methodology is based on the computation of harmonic coe$cients leading to e!ective estimates of the average ridge period. Moreover, since the mathematical de"nition of distance map completion is not unique, and is also ill-posed, a speci"c formulation was employed which minimizes value variations in the "nal distance map. Tests using the NIST SDB4 and other proprietary databases show the e!ectiveness of the approaches in about one-tenth of the overall computation time of a user identi"cation system. On the NIST SDB4 database, the recognition performance achieved by the two systems employing the geometric and spectral approaches are substantially the same and lead to the extraction of the same number of minutiv with respect to human experts.

References [1] B. Moayer, K.S. Fu, A syntactic approach to "ngerprint pattern recognition, Pattern Recognition 7 (1975) 1}23. [2] D.K. Isenor, S.G. Zaky, Fingerprint identi"cation using graph matching, Pattern Recognition 19 (2) (1986) 113}122. [3] F. Galton, Finger Prints, MacMillan, London, 1892. [4] Q. Xiao, H. Raafat, A combined statistical and structural approach for "ngerprint image postprocessing, Proceedings of the IEEE International Conference on Systems, Man and Cybernetics Conference, 1990, pp. 331}335. [5] The Science of Fingerprints: Classi"cation and Uses United States Department of Justice, Federal Bureau of Investigation, Washington, rev. 12}84, 1988. [6] W.-C. Lin, R.C. Dubes, A review of ridge counting in dermatoglyphics, Pattern Recognition 16 (1983) 1}8. [7] L. O'Gorman, J.V. Nickerson, An approach to "ngerprint "lter design, Pattern Recognition 22 (1989) 28}38.

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[8] D.C.D. Hung, Enhancement and feature puri"cation of Fingerprint Images, Pattern Recognition 26 (1993) 1661}1671. [9] C.I. Watson, C.L. Wilson, NIST Special Database 4, Fingerprint Database, National Institute of Standard and Technology, March 1992. [10] W.K. Pratt, Digital Image Processing, Wiley Interscience, New York, 1991. [11] J.S. Weszka, R.N. Nagel, A. Rosenfeld, A threshold selection technique, IEEE Trans. Comput. (1974) 1322-1326. [12] J.S. Weszka, A. Rosenfeld, Histogram modi"cation for threshold selection, IEEE Trans. Systems Man Cybernet. 9 (1979) 38}52. [13] N. Otsu, A threshold selection method form gray-level histograms, IEEE Trans. Systems Man Cybernet. 9 (1979) 62}66.

[14] J. Hadamard, La theH orie des eH quations aux deriveH es partielles, Editions Scienti"ques, Pekin, 1964. [15] A.N. Tikhonov, V.Y. Arsenin, Solution of Ill-Posed Problems, Winston and Wiley, Washington, 1977. [16] V.A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer, Berlin, 1984. [17] R. Courant, D. Hilbert, Methods of Mathematical Physics, Interscience Publisher, New York, 1953. [18] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes, Cambridge University Press, New York, 1990. [19] A. Farina, Zs. M. KovaH cs, A. Leone, Fingerprint minutiae extraction from skeletonized binary images, Pattern Recognition 32 (5) (1999) 877}889.

About the Author*ZSOLT M. KOVAD CS-VAJNA received his D. Eng. degree from the University of Bologna, Italy, in 1988. Since 1989 he has been with the Department of Electrical Engineering of the same University where he received a Ph.D. in Electrical Engineering and Computer Sciences in 1994 for his research on optical character recognition and circuit simulation techniques. He is currently Assistant Professor in Electronics. His research interests include pattern recognition (OCR, ICR, "ngerprint identi"cation), neural networks and circuit simulation techniques. He is a member of the Institute of Electrical and Electronics Engineers (IEEE), of the International Association for Pattern Recognition (IAPR-IC) and of the International Neural Network Society (INNS). About the Author*RICCARDO ROVATTI was born in Bologna, Italy, on 14 January 1969. He received the Dr. Eng degree (with honors) in Electronic Engineering and Ph.D. degree in Electronic Engineering and Computer Science from the University of Bologna, in 1992 and 1996, respectively. Since 1997 he has been a lecturer of Digital Electronics at the University of Bologna. He has authored or co-authored more than sixty international scienti"c publications. His research interest include fuzzy theory foundations, learning and CAD algorithms for fuzzy and neural systems, statistical pattern recognition, function approximation, non-linear system theory and identi"cation as well as applications of chaotic systems. About the Author*MIRKO FRAZZONI received the D. Eng. degree in Electrical Engineering in 1996 at the University of Bologna, Italy. He is presently with Praxis, UK.