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1、Extract from: D. Maltoni, D. Maio, A.K. Jain, S. Prabhakar Handbook of Fingerprint Recognition Springer, New York, 2003 3.6: Singularity and Core Detection (extract) (Copyright 2003, Springer Verlag. All rights Reserved.) 3 Fingerprint Analysis and Representation 96Figure 3.13. Segmentation of a fin
2、gerprint image as proposed by Ratha, Chen, and Jain (1995): a) original image; b) variance field; c) quality image derived from the variance field: a quality value “good,” “medium,” “poor” or “background” is assigned to each block according to its vari- ance; d) segmented image. Elsevier. 3.6 Singul
3、arity and Core Detection Most of the approaches proposed in the literature for singularity detection operate on the fin- gerprint orientation image. In the rest of this section, the main approaches are coarsely classi- fied and a subsection is dedicated to each family of algorithms. a) c) b) d) 3.6
4、Singularity and Core Detection 97 Poincar method index An elegant and practical method based on the Poincar index was proposed by Kawagoe and Tojo (1984). Let G be a vector field and C be a curve immersed in G; then the Poincar index PG,C is defined as the total rotation of the vectors of G along C
5、(see Figure 3.14). Figure 3.14. The Poincar index computed over a curve C immersed in a vector field G G. Let G be the field associated with a fingerprint orientation image1 D and let i,j be the posi- tion of the element ij in the orientation image; then the Poincar index PG,C(i,j) at i,j is com- pu
6、ted as follows. The curve C is a closed path defined as an ordered sequence of some elements of D, such that i,j is an internal point; PG,C(i,j) is computed by algebraically summing the orientation differences between adjacent elements of C. Summing orientation differences requires a direction (amon
7、g the two possible) to be associated at each orientation. A solution to this problem is to randomly select the direction of the first element and assign the direction closest to that of the previous element to each successive element. It is well known and can be easily shown that, on closed curves,
8、the Poincar index assumes only one of the dis- crete values: 0, 180, and 360. In the case of fingerprint singularities: () =region.singular typedelta a tobelongs , ifregionsingular typeloop a tobelongs , ifregionsingular type whorla tobelongs , ifregionsingular any tobelongnot does , if180-1803600ji
9、jijijij , iPC,G1 Note that a fingerprint orientation image is not a true vector field inasmuch as its elements are unori- ented directions. C v1 v2 v3 v4 v5 v6 v7 v8 v9 PG,C = - 263 v1 v2 v3 v4 v5 v6 v7 v8 v9 3 Fingerprint Analysis and Representation 98Figure 3.15 shows three portions of orientation
10、 images. The path defining C is the ordered sequence of the eight elements dk (k = 0.7) surrounding i,j. The direction of the elements dk is chosen as follows: d0 is directed upward; dk (k = 1.7) is directed so that the absolute value of the angle between dk and dk-1 is less than or equal to 90. The
11、 Poincar index is then com- puted as ()()() =+=708 mod 1 .kkkC,anglej , iPddG. Figure 3.15. Example of computation of the Poincar index in the 8-neighborhood of points be- longing (from the left to the right) to a whorl, loop, and delta singularity, respectively. Note that for the loop and delta exa
12、mples (center and right), the direction of d d0 is first chosen upward (to compute the angle between d d0 and d d1) and then successively downward (when computing the angle between d d7 and d d0). An example of singularities detected by the above method is shown in Figure 3.16.a. An interesting impl
13、ementation of the Poincar method for locating singular points was proposed by Bazen and Gerez (2002b): according to Greens theorem, a closed line integral over a vector field can be calculated as a surface integral over the rotation of this vector field; in practice, instead of summing angle differe
14、nces along a closed path, the authors compute the “rotation” of the orientation image (through a further differentiation) and then perform a local integration (sum) in a small neighborhood of each element. Bazen and Gerez (2002b) also provided a method for associating an orientation with each singul
15、arity; this is done by compar- ing the orientation image around each detected singular point with the orientation image of an ideal singularity of the same type. Singularity detection in noisy or low-quality fingerprints is difficult and the Poincar method may lead to the detection of false singular
16、ities (Figure 3.17). Regularizing the orienta- tion image through a local averaging, as discussed in Section 3.3, is often quite effective in preventing the detection of false singularities. PG,C(i,j) = 360 PG,C(i,j) = 180 PG,C(i,j) = -180 i,j d0 d1 d2 d3 d4 d5 d6 d7 i,j d0 d1 d2 d3 d4 d5 d6 d7 i,j d0 d1 d2 d3d4 d5 d6 d7 3.6 Singularity and Core Detection 99 Figure 3.1
