Statistical Methods for Automatic Interpretation of Digitally Scanned Fingerprints

Transcription

1 Statistical Methods for Automatic Interpretation of Digitally Scanned Fingerprints K.V. Mardia, A.J. Baczkowski, X. Feng, T.J. Hainsworth Department of Statistics, University of Leeds, Leeds LS2 9JT, U.K. Internal Report STAT 97/23 December 1997

2 Statistical Methods for Automatic Interpretation of Digitally Scanned Fingerprints K.V. Mardia, A.J. Baczkowski, X. Feng, T.J. Hainsworth Department of Statistics, University of Leeds, Leeds LS2 9JT, U.K. Internal Report STAT 97/23 December 1997 Abstract Orientation flow field is the starting point for feature extraction in various fingerprint algorithms. In recent papers, Jain and his colleagues (Ratha et al., 1995, Karu and Jain, 1995, Ratha et al. 1996) introduced a fundamentally new algorithm. Their main motivation for the algorithm has been to work at the problems from a fingerprint matching point of view. Our aim in contrast has been feature extraction from a dermatoglyphic point of view. Thus, we are not only interested in classification of the fingerprints but also other features such as ridge counts (number of ridges between any two points, say core points and tri-radius) and the average width of ridges in a region of fingerprint (see Loesch, 1983). Our end product is calculation of quantitative features rather than identification. Again, manual calculations of such quantities is a highly tedious task. We use some advanced techniques of spatial statistics, directional data analysis and Bayesian image analysis to obtain the orientation field. The algorithm is used to extract various features of interest from fingerprints. 1 Introduction This paper focuses on statistical techniques for extracting structural features from fingerprint images. We suppose that a fingerprint impression has been taken and is digitized to give an image, typically containing pixels, which can be then 1

3 processed or enhanced in some way to overcome the presence of noise which arises from an inadequate initial impression or a poor procedure. To improve fingerprint image quality, directional ridge enhancement is commonly employed. Fingerprint images can be considered as an oriented texture pattern with the lines on the image giving the ridges and valleys of the fingerprint. These lines flow in a locally constant direction; see for example the fingerprint image shown in Figure 1, see Section 2. This fingerprint is used in this paper for illustrative purposes. Computing this local direction at each point of the image defines the orientation field for the image. In this paper we use the spatial structure of the fingerprint image to provide information about the orientation field. This spatial structure is summarized using the semi-variogram, a basic tool of spatial statistics (see, for example, Cressie, 1991). The sample semi-variogram gives not only the local orientation field but also the distance between neighbouring lines in the image, and in turn the distance between the ridges or valleys of the fingerprint. To smooth the orientation field produced from this semi-variogram approach, we use techniques of directional statistics (see for example, Mardia, 1972). To further enhance the orientation field we use a Bayesian framework with a suitable prior for the directional field. For computation of the posterior mode, we used the iterative conditional mode procedure of Besag (1986). Most methods for computing orientation field are based on the variance of groups of pixels, for example, Mehtre (1993) used the intensity variance in nine directions while Sherlock et al. (1994) projected along 16 directions and chose the direction with maximum variance. There are other approaches; for example, Kawagoe and Tojo (1984) divided a fingerprint image into subregions and estimated the average direction of each subregion by counting micro-patterns of the image, while Karu and Jain (1996) used a method based on the grey-value sum of pixels. Our method appears to give better results than that of Karu and Jain (1996). To extract information about the ridges we use the mean grey level of blocks of pixels aligned with the orientation field at any location. Other techniques include, for example, thresholding to give a binary image, due to Mehtre (1993), but see also Ratha et al. (1995). The advantage of our method is that it does not require a large window about each pixel of interest since the semi-variogram provides the necessary distance 2

4 information. Using a classical thinning algorithm (Pavlidis, 1982, pp ), we thin the ridges to give a skeleton image with ridges having width of one pixel only. From this we extract count information about the ridges. Our ridge counting method gives very low bias and mean square error, superior to those based on using the grey-level of pixels. The orientation field can also be used to segment the fingerprint image into regions reflecting the quality of the original image. We do this using a moving window to assess the local pixel grey value variance where the window size depends upon the distance between local ridges estimated from the semi-variogram and does not need to be set manually as in the procedure of Ratha et al. (1995). We further use our orientation field for classification of the fingerprints by extracting feature points such as core and tri-radius points from the fingerprint. This application is supplemented by incorporating Karu and Jain s (1996) method to detect the flow pattern of the orientation field. The method involves the use of the Poincaré index, see Rosen (1970) and Kawagoe and Tojo (1984). 2 Orientation field obtained from semi-variogram Suppose that a fingerprint image is made up of a m n rectangular array of pixels. For a pixel located at (i, j) let its grey level value be x ij. A fingerprint image is then described by the array x = {x ij : row and column labels respectively. i = 1,, m and j = 1,, n}, where i and j are For each pixel (i, j), we obtain the sample semi-variogram along each of 16 equispaced directions making angle α d with the horizontal axis, where α d = πd/16 for d = 0,, 15. To ensure that the calculated sample semi-variogram reflects the local spatial structure about each pixel we only include a maximum of (2r + 1) pixel grey level values centred on (i, j) in the semi-variogram. Typically we used r 30. The sample semi-variogram at lag h and direction d centred on pixel (i, j) is given by, g ij (h; d) = 1 2(2r + 1 h) r h k= r (y k y k+h ) 2, where y k denotes the grey level value at a distance k from (i, j) along line d. Thus 3

5 for d = 0, y k = x ik. For an arbitrary direction d, y k is assigned the grey level value of the pixel at the corresponding position. To ensure that g ij (h; d) is averaged over a sufficient number of terms we only consider lags h up to a maximum of 40 in this work. Figure 1 shows a blurred fingerprint which will be used for illustrative purposes throughout this paper. Figure 2 shows the semi-variogram functions plotted against lag h for a selection of directions d for a typical pixel (i, j) located in the fingerprint image of Figure 1. INSERT FIGURE 1. INSERT FIGURE 2. It is observed that as the direction d approaches the direction orthogonal to the ridge direction, the semi-variogram plots become more cyclical and the average distances between any two neighbouring local minima become smaller. This observation is easily explained. Suppose that the ridges can be represented by parallel lines a common distance H apart. For a direction making angle θ with the ridge lines, the distance between intersections of neighbouring ridges is H cosecθ which is minimized for θ = 1 π. At lags h H cosecθ the semi-variogram will be approximately 2 zero, and at lags h 1 H cosecθ the semi-variogram will be averaged over many pairs 2 of pixels with one on a ridge and one in a valley which gives a large semi-variogram value. The algorithm for obtaining both the local direction of the orientation field and the distances between ridges from the semi-variogram function is summarized as follows for pixel (i, j). (1) Find the local minima of the semi-variograms in each direction d. These can be found from the sample semi-variogram g ij (h; d) by comparison with the adjacent values g ij (h + 1; d) and g ij (h 1; d). If n ij (d) minima are found, let H ij (k, d) denote the distance between the (k 1)th and kth minima, k = 2,, n ij (d). To ensure that a local minima of the semi-variogram has been found we also check g ij (h; d) with the values g ij (h ± 2; d) and g ij (h ± 3; d). Furthermore, if we find a case with H ij (k, d) < 5 then we replace the two apparent minima by one at their average location; this is done to avoid spurious minima 4

6 since the minimum distance between the ridges in the fingerprint images we have used is greater than 5 pixels. (2) Calculate the mean distance H ij (d) between the pairs of neighbouring local minima for each direction d. If only 0 or 1 local minima were found then H ij (d) is not defined. (3) If there is some direction d giving a unique minimum mean, Hij (d ), then this direction is perpendicular to the local ridge orientation and H ij (d ) gives the distance between ridges in the neighbourhood of (i, j). If there are several directions, d D say, giving the same mean value H ij (d) for all d D then proceed to step (4). If H ij (d) is not defined for all directions d then the orientation of the local ridges cannot be determined. In practice this does not occur, there being at least one direction d for which H ij (d) exists so that a minimum can be found. (4) Assess the cyclical behaviour of the semi-variogram function for each direction d D by evaluating for each such direction d the sample variance c ij (d) of the distances H ij (k, d). Select the direction d which minimizes c ij (d) for d D. This direction gives the direction perpendicular to the local ridge direction. As before H ij (d ) gives the local distance between neighbouring ridges. Given this orthogonal direction d it is trivial to obtain the local direction d ij of the orientation field at pixel (i, j). 3 Smoothing the orientation field The directions obtained from the semi-variogram approach are usually quite noisy and the directions for some pixels cannot be estimated. Therefore they need to be smoothed and averaged in a local neighbourhood. A method which is similar to that of Karu and Jain (1995) and Wilson et al. (1993) is used but using a proper mean from the statistical theory of directional data (see for example, Mardia, 1972). For a pixel located at (i, j) let the computed orientation field have local direction d ij with corresponding angle α ij, α ij [0, π). 5

7 To smooth the directional image, we present the direction α ij as vector. Since the angles α ij and π + α ij are not distinguished, this vector is not directed. We would describe α ij as axial data, see Mardia (1972, p1). The directed vector for direction α ij is γ ij = (cos 2α ij, sin 2α ij ), where cos 2α ij and sin 2α ij are the Cartesian co-ordinates of the directed vector γ ij. Karu and Jain (1996) seem to have used this approach. The directional image can now be smoothed by averaging the two components in γ ij separately. We used a 3 3 mean box filter (see Pavlidis, 1982, Section 3.4). The smoothed directional vector becomes, γ ij = ( cos 2α k, 1 ) 9 sin 2α k, 9 k=1 k=1 where α k denotes the directions of the pixels in the 3 3 box centred at (i, j). The smoothed direction ᾱ ij can then be found from γ ij = (cos 2ᾱ ij, sin 2ᾱ ij ), see Mardia (1972, p26). The general angle ᾱ ij is discretized to give a smoothed direction d ij taking sixteen equi-spaced possible values. Repeating these calculations for every pixel gives a set of smoothed directions d for a fingerprint image, which we refer to as the smoothed orientation field. An alternative method for obtaining the orientation field is the grey-value sum method of Karu and Jain (1995) and Wilson et al. (1993), following the work of Stock and Swonger (1969). This method uses a 9 9 mask centred on the pixel of interest and adds up the grey level value of selected pixels in eight different directions to obtain the orientation direction. We found that this procedure does not compare well in practice to the semi-variogram method. Furthermore it does not readily give the distance between adjacent ridges. Also, as discussed in Section 4, our method can be improved further which is especially important for noisy fingerprint images as shown in Figure 3. Example: For the fingerprint image shown in Figure 1 the smoothed orientation field is obtained using the semi-variogram method and the method of Karu and Jain. These are shown superimposed on the original fingerprint image in Figures 3a and 3b respectively. INSERT FIGURE 3. It can be seen that the semi-variogram method gives a more accurate orientation field than the method of Karu and Jain. In clear areas, such as in the top part of the 6

8 image, both methods satisfactorily give the directions of the ridge flows. In blurred areas, such as the middle part of the image, the directions obtained from the semivariogram method more precisely express the ridge flows than the procedure of Karu and Jain. In blurred regions the semi-variogram is superior because it extracts information about the orientation field over a wider region than the 9 9 mask of Karu and Jain. Also, note that the letter C, used for identification purposes, goes through the same directional process and it is not surprising to see degradation by both methods. Figure 4a shows a small area of the fingerprint image containing two sections of ridges. The distance between the ridges is approximately 11 pixels. INSERT FIGURE 4. Figure 4b shows the estimated ridge spacing obtained from the semi-variogram method for the same image. It can be seen that at all pixels the correct distance is obtained. A 3 3 mean filter is used to smooth these distances and yields a value of 11 in nearly all the pixels in Figure 4. 4 Bayesian framework for improving the orientation field To further enhance the orientation field we use a Bayesian framework with a suitable prior for the directional field. The computational work involved for the maximum a priori (MAP) solution is intense since the images are usually very large-scale data. The iterated conditional modes (ICM) method of Besag (1986) is used to reduce the computational burden. Suppose the observed fingerprint image x is a realization of a random matrix, X = {X ij : i = 1,, m and j = 1,, n}. Let an arbitrary orientation field be denoted by d = {d ij : i = 1,, m and j = 1,, n}, where d ij is an integer between 0 and 15 indicating the corresponding direction for pixel (i, j). This can be interpreted as a realization of a random matrix, D = {D ij : i = 1,, m and j = 1,, n}, 7

9 where D ij assigns a direction to pixel (i, j). We make the following two standard assumptions. Assumption 1. Given any orientation field d, the random variables X ij are conditionally independent and each X ij has the same known conditional density function f(x ij d ij ). Thus, the conditional likelihood of the observed fingerprint image x given d is, l(x d) = m n f(x ij d ij ). i=1 j=1 Assumption 2. The true orientation field d is a realization of a locally dependent Markov random field with specified distribution p(d). The conditional density f(x ij d ij ) is taken to be Gaussian with mean µ ij and variance σij 2, where the variance is estimated by the sample variance s2 ij of the nine grey levels for pixels in the 9 9 window centred on (i, j). The mean µ ij is estimated by x ij (d ij ), the grey level mean of those pixels within the 9 9 window which lie along direction d ij. The prior distribution p(d) is taken to be from the Ising model (see Besag, 1986), m n p(d) exp {βn(d ij )}, i=1 j=1 where n(d ij ) is the number of the pixels located in the surrounding 3 3 block and having the same direction as pixel (i, j). The motivation for this is that we expect the directions of neighbouring pixels in an image to be similar. The parameter β is taken to be positive to encourage the directional similarity of neighbouring pixels. We have used β = 1.5 which seems to work well. Taking β = 0 gives the maximum likelihood estimate of the orientation field which gives the direction with pixels having mean grey level closest to the corresponding observed pixel. As β, the ICM provides a smoothing solution acting on the initial orientation field. The posterior density p(d x) of the orientation field d, given the observed fingerprint image x is, from Bayes s theorem, p(d x) = l(x d) p(d). 8

10 We would estimate the orientation field using the value ˆd which maximizes this posterior density. Since the fingerprint image size is large this imposes considerable computational demands. To overcome this problem we use the iterative conditional mode procedure of Besag (1986). If d denotes a provisional estimate of the orientation field at all locations excluding pixel (i, j) then we can estimate d ij by maximizing p(d ij x, d) with respect to d ij, where, p(d ij x, d) f(x ij d ij )p(d ij d). On taking logarithms it can be seen that this is achieved by minimizing, {x ij x ij (d ij )} 2 /2s 2 ij βn(d ij ). When applied to each pixel in turn, this procedure defines a single cycle of an iterative algorithm for estimating the orientation field. As an initial estimate d we use the results of the semi-variogram method. To obtain the estimated orientation field ˆd we apply the algorithm for a fixed number of cycles or until convergence occurs. In practice convergence is rapid with few changes occurring after about six cycles. For the smoothed orientation field of Figure 3a the resultant orientation field after using the ICM method is shown in Figure 5, which shows the orientation field superimposed on the original fingerprint image. Comparison of Figure 3a and Figure 5 shows a clear improvement from using ICM. INSERT FIGURE 5. To show the improvement clearly, three regions in Figure 3a and the corresponding regions in Figure 5 are enlarged and shown in Figure 6. INSERT FIGURE 6. The small squares and triangles in Figures 5 and 6 show the core and delta points for the fingerprint patterns. Their recognition from the orientation field is discussed in detail in Section 7. We note here that the positions of these points in the ICM output orientation field are more accurate than using the original orientation field. 9

11 5 Extraction of ridges A commonly used technique for extracting ridges is to threshold the image to obtain a binary image, see Mehtre (1993). This thresholding technique uses a large window to ensure that at least one ridge and one valley is included in a window at every pixel. The orientation field has been used by some workers to extract the ridges from a fingerprint image. For example, Ratha et al. (1995) mapped an image into a waveform whose values are the sums of the grey values of the image pixels along the directions orthogonal to the directions of pixels. The method introduced here is similar in spirit to their method except that we use the information about the ridge distance. For each pixel (i, j) in the fingerprint image let the corresponding smoothed orientation direction after ICM be d ij and let d ij be the perpendicular direction. Let H ij (d ij ) denote the average distance between ridges at pixel (i, j), obtained from the semi-variogram method of Section 2. Define the directed mean of pixel (i, j) along direction d ij as the average grey level value x ij of the p pixels on each side of (i, j) along direction d ij and including the value x ij. Thus x ij is the average of (2p + 1) grey level values, x ij = 1 (2p + 1) p k= p where y k denotes the grey level value at a distance k from (i, j) along line d ij. Repeating the calculation of x ij for all pixels gives the directed mean image x along the directions of the orientation field of the original fingerprint image. Figure 7b shows this directed mean image for the fingerprint image shown in Figure 7a. INSERT FIGURE 7. This directed mean image can be used to extract partial below) from a fingerprint image. y k, ridges (as defined To do this we consider the directional means x k of the H ij (d ij ) pixels a distance k away from pixel (i, j) along direction d ij, for k = 1 2 H ij (d ij ),, 1, +1,, 1 2 H ij (d ij ), so excluding x ij. Then we classify pixel (i, j) as a ridge or valley (background) according to the rule, 0 (ridge) if x ij max x k, b ij = 1 (valley) otherwise. 10

12 The extracted ridges are assigned a value 0 (bright) and the other pixels are assigned a value 1 (dark background). The results of this procedure are shown in Figure 7c. This shows one advantage of this approach in that only the central part of the ridges are extracted, thus achieving a partial thinning of the ridge. The binary ridge image is further smoothed prior to obtaining a smooth skeleton. Ratha et al. (1995) used a mask along the orientation field to smooth the extracted ridges, the size of the mask being determined empirically. Similarly we use a 1 9 mask centred on pixel (i, j) and orientated along direction d ij. If more than a quarter of the pixels have been assigned as ridge then pixel (i, j) is set to be ridge, otherwise it is set as valley. To obtain a one-pixel width skeleton of the ridge, the thinning procedure of Pavlidis (1982, Section 9.2) is used here. Figure 7d shows the smoothed binary ridge image and Figure 7e shows the corresponding ridge skeleton. Aberrations in the smoothed binary ridge image can have an adverse influence on a subsequently derived ridge skeleton, giving spikes which lead to spurious ridge bifurcations and endings. Wrongly assigned ridge pixels can lead to holes in the extracted ridge skeleton. These problem are eliminated using a row-column filling algorithm which replaces a binary value b ij = 0 by 1 if pixel (i, j) has neighbours (i ± 1, j) or (i, j ± 1) which have binary value 1. Repeating this over all pixels gives a new binary image. Using the thinning algorithm now gives the final ridge skeleton shown in Figure 7f. The problems visible in Figure 7e have been successfully cleaned up by the filling algorithm. A finger ridge count gives the number of ridges crossing an imaginary line between any two points on a fingerprint image. It is an important dermatoglyphic measurements and is used to indicate the fingertip size and the ridge density in a given region. The ridges containing the points are both excluded from the count. If a ridge bifurcates before or on meeting the line, two ridges are counted. Further details of ridge counting are contained in Schaumann and Alter (1976, Chapter 3). This procedure is summarized in Figure 8. INSERT FIGURE 8. A simple ridge counting threshold algorithm counts the number of times a pixel grey level value along the line connecting the two points is less than a threshold when the grey level value of the previous pixel is greater than or equal to the threshold. The 11

13 threshold is given by, 1 2 (x max x min ) where x max is the largest grey level value along the line and x min is the smallest. To compare the counting results of this grey level threshold algorithm with those from a ridge skeleton image, 30 pairs of points are randomly selected on the image. Since a one-pixel wide ridge along a diagonal direction may not provide a clear intersection with an arbitrary line also one pixel wide (due to th0e nature of the square pixels) the ridge skeletons are expanded to be of width three pixels. For each line k, k = 1,, 30, the number, say n 1k, of ridge crossings is found for the expanded ridge skeleton image, and the number, say n 2k, for the corresponding grey-level fingerprint image. The true ridge number, say n k, is determined visually using the original grey-level fingerprint image. The bias and mean square error of the two methods are estimated by the mean of (n ik n k ) and (n ik n k ) 2 respectively for i = 1, 2. Table 1 gives summary statistics for the two methods averaged over the 30 lines. Table 1: Summary Statistics of Ridge Count Methods. Bias Mean Square Error Ridge skeleton method Grey-level threshold algorithm It can be seen from Table 1 that using the ridge skeleton method gives both a smaller bias and smaller mean square error than using grey-level thresholding. Furthermore, we find that for the grey level thresholding technique, the errors increase as we increase the line length along which the ridge count takes place, whereas using the ridge count skeleton gives low error rates independent of line length. The method of Lin and Dubes (1983) used thresholding and a ridge counter based on the Hough transform of the binary image. Our method seems be to superior in that the counting algorithm is simple, based on the ridge skeletons which express ridge shape and are not sensitive to errors. Furthermore, our method readily extends to ridge counting on palm prints. 12

14 6 Segmentation into different quality regions Segmentation identifies areas of an image that appear uniform, and subdivides the image into regions of uniform appearance. Segmentation highlights those regions which are noisy or which have good image quality. Ratha et al. (1995) used the variance of grey level values in a direction orthogonal to the orientation field to segment a fingerprint image into different quality regions. The problem with their method is that the window size used to select pixels had to be chosen manually. If the window size is smaller than the width of a ridge of interest then the whole block of pixels is located on a ridge and the variance will be small, even in the direction orthogonal to the ridges. Our approach overcomes this problem by using information about the distance between local ridges obtained from the semi-variogram method to define the window size. For pixel (i, j) we consider the H ij (d ij ) + 1 pixels a distance k away from pixel (i, j) along direction d ij orthogonal to the orientation field, for k = 1 2 H ij (d ij ),, 1 2 H ij (d ij ). The sample variance of the grey levels of these pixels we denote s 2 ij (d ij ) and refer to as the adaptive variance at pixel (i, j). They give variances in a direction orthogonal to the ridge direction. Repeating this calculation at every pixel in the fingerprint image gives the adaptive variance field. The background areas and noisy regions in a fingerprint image have no directional dependence, thus the values of the adaptive variances are randomly arranged, whereas clear regions with high image quality exhibit a large variance in the direction orthogonal to the orientation field and very small variance along the ridges. Hence we can use the adaptive variance field to assess the quality of the fingerprint image. To remove noise and obtain smoothed quality regions we can use a 9 9 moving average filter. For the fingerprint image in Figure 1 we show the smoothed adaptive variance field in Figure 9a. INSERT FIGURE 9. A large adaptive variance indicates a good quality region and a small adaptive variance indicates a poor quality region. For the adaptive variance field shown in Figure 9a, the maximum and minimum 13

15 smoothed adaptive variances are s 2 max = and s 2 min = 3.60, respectively. An efficient classifications into four quality regions can be made by dividing the interval (s 2 min, s2 max) into four subintervals denoting background, poor, middle, and good quality respectively, with corresponding assigned grey levels 0, 70, 130 and 255. The resulting quality field is shown in Figure 9b in which the dark regions indicate good quality. An alternative method for segmenting a fingerprint image into different quality regions may be based on the global grey level values of pixels. The global mean x and global variance s 2 of all pixels in a fingerprint image are first calculated. The fingerprint image is then divided into subregions (for example, block of 9 9 pixels making up each subregion) and the local mean x u of pixels in subregion u is calculated. If x u x ± 2s then subregion u is regarded as a good quality region, otherwise it is regarded as having bad quality. While this procedure successfully detects the background elsewhere in the image, it is not as efficient as the adaptive variance approach described above. The segmentation into different quality region is useful in further analysis of a fingerprint image. For example, a type of fingerprint pattern recognition is based on the ridge details, see Ratha et al. (1995) and Schaumann and Alter (1976, Chapter 3). With knowledge about the quality of each part of a fingerprint image, the ridge details can be weighted according to the quality of their location. 7 Classification of fingerprints using orientation fields The classification of fingerprints is usually done at two different levels. Although the ridges can be viewed as approximately locally parallel lines, detailed inspection of the ridge patterns reveals numerous irregularities of directions, discontinuities, and branching of individual ridges. These intricate details of ridge structure, called minutiae, are highly variable and their characteristics are unique to an individual. They give a tool for personal identification, see Ratha et al. (1995). More broadly, Galton (1892) divided fingertip ridge patterns into three groups: arches, loops, and whorls. Numerous sub-classifications have been subsequently defined. For example, Schaumann and Alter (1976, Chapter 3) divided the fingertip patterns into eight types: simple arch, tented arch, ulnar loop, radial loop, simple whorl, central pocket whorl, double loop whorl, and accidental whorl, while Kawagoe and Tojo (1984) 14

16 used the classifications left loop, right loop, left pocket, right pocket, simple and twin whorl, simple and tented arch, and irregular. In the method described in this section we extract feature points in a fingerprint image after smoothing the orientation field. We use the the position and the number of the extracted feature points to classify the fingerprints into six basic patterns: simple arch, tented arch, left loop, right loop, simple whorl, and twin loop. There are three types of directional flow for the orientation field in a neighbourhood around any pixel, as shown in Figure 10. INSERT FIGURE 10. The centre points in the different subregions are called normal, core and delta points respectively. The core point and the delta point are often referred to as singular points. Figure 11 shows schematic drawings of the six fingerprint patterns together with the associated core and delta points, shown as squares and triangles respectively. INSERT FIGURE 11. To detect the singular points in an orientation field, the Poincaré index is calculated for each subregion, see for example Kawagoe and Tojo (1984) and Rosen (1970). We adopt the method of Karu and Jain (1996) to calculate the Poincaré index at any point. Having found the core and delta points we can use their number to classify the fingerprint into one of three categories: (1) arch with 0 core and 0 delta points, (2) tented arch, left loop or right loop with 1 core and 1 delta point, (3) whorl or twin loop with 2 core and 2 delta points. To discriminate between a tented arch and a loop we consider the line drawn between the core and delta points. Let the line have angle β, with zero denoting the horizontal axis. In a tented arch, the direction of the connecting line is along the local direction vector of the orientation field, while in a loop the line intersects the local direction vector transversely. Let α i, for i = 1,, m, be the local direction angles along the line 15

17 segment. If the averaged sum, S = 1 m sin(α i β), m i=1 is less than a threshold (here taken to be 0.2), then the image is classified as a tented arch, otherwise it is a loop. See Figure 11b, 11c, 11d. To discriminate between a left and right loop we denote the core point by c and the delta point as D, see Figure 11c an 11d. Starting at C we can follow the direction vectors from one pixel to the next until meeting the boundary of the image at point B. Suppose the coordinates of C, D and B are (C x, C y ), (D x, D y ) and (B x, B y ) respectively, then the image is classified as a right loop if D x C x D y C y > B x C x B y C y, and as a left loop otherwise. We can discriminate between a whorl and a twin loop by connecting the two core points with a straight line, see Figures 11e and 11f. If α i, for i = 1,, m, are the local direction angles on this line, and the averaged sum, S = 1 m sin(α i β) m i=1 is less than a threshold (again taken to be 0.2), then the image is classified as a whorl, otherwise it is a twin loop. This methodology works successfully providing the singular points are not located too close to the image boundaries. The results may be improved by careful consideration of the boundary process. Kawagoe and Tojo (1984) used the Poincaré index to detect four types of flow pattern in an orientation field; core, normal, delta, and whorl. While the method used in this paper only detects the first three patterns, we can overcome this by computing the Poincaré index along a 3 3 rectangle centred at the pixel of interest. 16

18 8 Discussion In summary we have introduced a powerful new technique to interpret automatically fingerprints for dermatoglyphic application. The technique extends easily for full handprints. Our semi-variogram method takes about 5 seconds to process one fingerprint image on a SUN SPARC 20 work-station. This is comparable with the method of Karu and Jain (1996) which also takes about 5 seconds. We have shown that the semi-variogram provides a useful tool in constructing the orientation field. We also show how this orientation field can be smoothed using directional statistics, and enhanced using a Bayesian framework. Various dermatoglyphic features can be then computed. existing methods. We compare and contrast our method with other A shortened version of this report is given in Mardia et al. (1997). References [1] Besag, J. (1986). On the statistical analysis of dirty pictures. Journal of Royal Statistical Society, B 48, [2] Cressie, N. A. C. (1991). Statistics for Spatial Data. Wiley, New York. [3] Galton, F. (1892). Finger Prints. Macmillan, London. [4] Karu K. and Jain A.K. (1996). Fingerprint classification. Pattern Recognition, 29, [5] Kawagoe, M. and Tojo, A. (1984). Fingerprint pattern classification. Pattern Recognition, 17, [6] Loesch, L. (1983). Quantitative Dermatoglyphics. Oxford University Press. [7] Mardia, K.V. (1972) Statistics of Directional Data. Academic Press, London. [8] Mardia, K.V., Baczkowski, A.J., Feng, X., and Hainsworth, T.J. (1997) Statistical methods for automatic interpretation of digitally scanned finger prints. To appear Pattern Recognition Letters. 17

19 [9] Mehtre, B. M. (1993). Fingerprint image analysis for automatic identification. Machine Vision and Applications, 6, [10] Pavlidis, T. (1982). Algorithms for Graphics and Image Processing. Computer Science Press, New York. [11] Ratha, N.K., Chen, S. and Jain, A.K. (1995) Adaptive flow orientation based feature extraction in fingerprint images. Pattern Recognition, 28, [12] Ratha, N.K., Karu, K., Chen, S. and Jain, A.K. (1996). A real-time matching system for large fingerprint databases. IEEE-PAMI. 18, [13] Schaumann, B. and Alter,M. (1976). Dermatoglyphics in Medical Disorders. Springer-Verlag, New York. [14] Sherlock, B.G., Monro, D.M. and Millard, K. (1994). Fingerprint enhancement by directional Fourier filtering. IEE Proc. Vis. Image Signal Processing, 141, [15] Stock, R. M. and Swonger, C. W. (1969). Development and evaluation of a reader of fingerprint minutiae. Cornell Aeronautical Laboratory, Technical Report CAL No. XM-2478-X-1: [16] Wilson, C.L., Candela, G.T. and Watson, C.I. (1993). Neural network fingerprint classification. Journal of Artificial Neural Networks, 1,

20 Figure 1: Original image (from Schaumann and Alter, 1976, p30). 19

21 d=0 d=4 Semi-variogram Semi-variogram h h d=8 d=12 Semi-variogram Semi-variogram h h Figure 2: Representative semi-variograms plotted against lag h for various directions d for part of the image in Figure 1. 20

22 Figure 3: Orientation fields superimposed on finger print image. Top a: using semivariogram method. Bottom b: using method of Karu and Jain. 21

23 Figure 4: Estimated ridge distances. Top a: a small part of a fingerprint image so no edge effects. Bottom b: estimated ridge distance at each pixel using semi-variogram method. 22

24 Figure 5: Fingerprint image and orientation field produced by ICM method. 23

25 Figure 6: Left a: fingerprint image and smoothed orientation field for three regions of Figure 3. Right b: corresponding results after ICM. 24

26 Figure 7: Extracting ridges from a fingerprint image. Top left a: part of fingerprint image. Top right b: directed mean image. Centre left c: extracted binary ridge image. Centre right d: smoothed binary ridge image. Lower left e: ridge skeleton. Lower right f: ridge skeleton after filling algorithm and thinning used. 25

27 Original fingerprint image Obtain the orientation field and the distances between ridges using semi-variogram Directional data analysis for smoothing the directions Obtain the directional mean image Improved directions by ICM Extract partial ridges Smooth the extracted ridges using the orientation field Thin to obtain one-pixel-width ridge skeletons Expand ridge skeletons into three-pixel-width Ridge count on the expanded skeleton image Figure 8: Block diagram of the ridge counting process. 26

28 Figure 9: Left a: smoothed adaptive variance field. Right b: image segmented into regions of different quality (dark region = good quality). Figure 10: Three types of neighbourhood directional flow. The circle indicates the centre point in each type. Left a: normal. Centre b: core. Right c: delta. 27

29 C b D C C B β D D β B β β Figure 11: Six different fingerprint patterns showing core points (solid squares) and delta points (solid triangles). Top left a: simple arch. Top right b: tented arch. Centre left c: left loop. Centre right d: right loop. Bottom left e: simple whorl. Bottom right f: twin whorl 28