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1、Extract from: D. Maltoni, D. Maio, A.K. Jain, S. Prabhakar Handbook of Fingerprint Recognition Springer, New York, 2003 4.3: Minutiae-based Methods (extract) (Copyright 2003, Springer Verlag. All rights Reserved.) 4.3 Minutiae-based Methods 141Makekau (1991), Grycewicz (1995, 1996), Rodolfo, Rajbenb
2、ach, and Huignard (1995), Grycewicz and Javidi (1996), Petillot, Guibert, and de Bougrenet (1996), Soifer et al. (1996), Gamble, Frye, and Grieser (1992), Wilson, Watson, and Paek (1997), Kobayashi and Toyoda (1999), and Watson, Grother, and Casasent (2000). However, these optical systems usually su
3、ffer from rotation and distortion variations and the hardware/optical components are complex and expensive; therefore, optical fingerprint matching technology has not reached satisfactory maturity yet. 4.3 Minutiae-based Methods Minutiae matching is certainly the most well-known and widely used meth
4、od for fingerprint matching, thanks to its strict analogy with the way forensic experts compare fingerprints and its acceptance as a proof of identity in the courts of law in almost all countries. Problem formulation Let T and I be the representation of the template and input fingerprint, respective
5、ly. Unlike in correlation-based techniques, where the fingerprint representation coincides with the finger- print image, here the representation is a feature vector (of variable length) whose elements are the fingerprint minutiae. Each minutia may be described by a number of attributes, including it
6、s location in the fingerprint image, orientation, type (e.g., ridge termination or ridge bifurca- tion), a weight based on the quality of the fingerprint image in the neighborhood of the minu- tia, and so on. Most common minutiae matching algorithms consider each minutia as a triplet m = x,y, that i
7、ndicates the x,y minutia location coordinates and the minutia angle : ,n.j,y,x,.,m.i,y,x,.,jjjjniiiim1 1 2121 = mmmmImmmmTwhere m and n denote the number of minutiae in T and I, respectively. A minutia jmin I and a minutia im in T are considered “matching,” if the spatial dis-tance (sd) between them
8、 is smaller than a given tolerance r0 and the direction difference (dd) between them is smaller than an angular tolerance 0: ()()(),ijijijryyxx,sd022+= mm and (5) ()()0360 min=ijijij,ddmm. (6) Equation (6) takes the minimum of ij and ij360 because of the circularity of angles (the difference between
9、 angles of 2 and 358 is only 4). The tolerance boxes (or hy-4 Fingerprint Matching 142per-spheres) defined by r0 and 0 are necessary to compensate for the unavoidable errors made by feature extraction algorithms and to account for the small plastic distortions that cause the minutiae positions to ch
10、ange. Aligning the two fingerprints is a mandatory step in order to maximize the number of matching minutiae. Correctly aligning two fingerprints certainly requires displacement (in x and y) and rotation () to be recovered, and likely involves other geometrical transformations: scale has to be consi
11、dered when the resolution of the two fingerprints may vary (e.g., the two fingerprint images have been taken by scanners operating at different resolu- tions); other distortion-tolerant geometrical transformations could be useful to match minu- tiae in case one or both of the fingerprints is affecte
12、d by severe distortions. In any case, tolerating a higher number of transformations results in additional degrees of freedom to the minutiae matcher: when a matcher is designed, this issue needs to be carefully evaluated, as each degree of freedom results in a huge number of new possible alignments
13、which significantly increases the chance of incorrectly matching two fingerprints from differ- ent fingers. Let map(.) be the function that maps a minutia jm (from I) into jm according to a given geometrical transformation; for example, by considering a displacement of x, y and a counterclockwise ro
14、tation around the origin1: ()+ = =jjjjjjjj,y, x,y,x,y,xmapmm, where + = yx yx yxjjjj cossinsincos. Let mm(.) be an indicator function that returns 1 in the case where the minutiae jm and im match according to Equations (5) and (6): ()()() = otherwise.0and 100ijij ij,ddr,sd,mmmmmmmm Then, the matchin
15、g problem can be formulated as ( )()() =miiiP,y, xP,y, x,mapmm1maximizemm, (7) where P(i) is an unknown function that determines the pairing between I and T minutiae; in particular, each minutia has either exactly one mate in the other fingerprint or has no mate at all: 1 The origin is usually selected as the minutiae centroid (i.e., the average point); before the matching step, minutiae coordinates are adjusted by subtracting the centroid coordinates. 4.3 Minutiae-based Methods 1431. P(i) = j indicates that the mate of the mi
