Q = � (q 1 x, q 1 y, β 1 ), ....., (q Q x , q Q y , β Q ) � . (A.35) where |P| = P , |Q| = Q, and (p i x , pi y , αi ) are the three features associated with the i th minutiae in set P. The basic assumption made here is that the second point set Q is a rotated, scaled, and translated version <strong>of</strong> the first set P, where points may be shifted by random noise, some points may be added and some points deleted. The task <strong>of</strong> fingerprint registration is to recover this unknown transformation. Since we do not know whether the two fingerprints are the same or not (i.e., images <strong>of</strong> the same finger), we attempt to find the ’best’ transformation in the sense that when applying the transformation to the minutiae points <strong>of</strong> the set P, as many <strong>of</strong> these points as possible overlap with the minutiae points from the set Q. Two overlapping points are considered as a match only if they have the same direction. There may be minutiae points in either set that do not match with any point in the other set. The matching score is computed for each transformation after discretizing the set <strong>of</strong> all allowed transformations. Consider a transformation Fs,θ,∆x,∆y : R 2 → R 2 , given by, ⎛ ⎜ Fs,θ,∆x,∆y ⎝ x ⎞ ⎛ ⎞ ⎛ ⎟ ⎜ ⎠ = s ⎝ y cosθ −sinθ sinθ ⎟ ⎜ ⎠ ⎝ cosθ x ⎞ ⎛ ⎟ ⎜ ⎠ + ⎝ y ∆x ⎞ ⎟ ⎠ , ∆y (A.36) where s, θ, and (∆x, ∆y) are the scale, rotation, and shift parameters, respectively. The space <strong>of</strong> transformations consists <strong>of</strong> quadruples (s, θ, ∆x, ∆y), where each pa- rameter is discretized into a finite set <strong>of</strong> values: s ∈ s1, .., sK , θ ∈ θ1, ..., θL, ∆x ∈ ∆x1, ..., ∆xM and ∆y ∈ ∆y1, ..., ∆yN Matching scores for the transformations are collected in the accumulator array A, where the entry A(k, l, m, n) counts the evidence for the transformation Fsk,θl,∆xm,∆yn. The array A is filled as follows: For each pair (p, q), where p = (pi x , piy ) is a point in the set P and q = (q j x, q j y) is a point in the set Q, find all possible transformations that map p to q and increment the evidence for these evidence for these transformations in array A. For every pair <strong>of</strong> values, (sk, θl), there is exactly one shift vector (∆x, ∆y) t such that Fsk,θl,∆x,∆y(p) = q, and it can be obtained as, ⎛ ⎜ ⎝ ∆x ⎞ ⎛ ⎟ ⎜ ⎠ = q − sk ⎝ ∆y cosθl ⎞ −sinθl sinθl ⎟ ⎠ p. cosθl (A.37) 140
Paired Minutiae Minutiae with unmatched angle Unmatched Minutiae Unmatched Minutiae (a) (b) (c) (d) Query fingerprint Minutia Database fingerprint minutia Bounding box Figure A.17: Various cases in minutiae matching. (a) Matched Minutiae; (b) Minutiae with unmatched angle; (c) and (d) Unmatched minutia. The values ∆x and ∆y need to be quantized to the nearest bins corresponding to ∆xm and ∆yn. A.6.2 Minutia Pairing The minutiae need to be paired or matched if their component or features (x, y, α) are equal within some tolerance after registration. For minutia pairing, we use a bounding box (also called tolerance box) <strong>of</strong> specific width and angle tolerance in order to accommodate the shift in the minutia angle attribute. One bounding box is drawn for every minutiae <strong>of</strong> query fingerprint image to search for it’s match. We can obtain four possible cases during matching: (i) Minutiae from both fingerprints match in all components, hence denoted by paired minutiae (Fig. A.17(a)), (ii) The minutia from query fingerprint matches with minutia from database in x and y coordinate, but direction remain unmatched. These minutiae are called minutiae with unmatched angle (Fig. A.17(b)), (iii) No minutia from database fingerprint falls within bounding box <strong>of</strong> the minutia from query image (Fig. A.17(c)) and (iv) No minutia from database fingerprint matches with the minutia from query image (Fig. A.17(d)). In last two cases, the minutia from query image is denoted by unmatched minutia. In our case, bounding box size is set to 50. Angle tolerance is set to 0.1 radians. 141
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DEVELOPMENT OF EFFICIENT METHODS FO
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To My Parents, Sisters and Dearest
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Mirnalinee, Shreyasee, Lalit, Manis
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LDC Linear Discriminant Classifier
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3 An Efficient Method of Face Recog
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A.6 Minutiae Matching . . . . . . .
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CHAPTER 2 Literature Review This re
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PCA Projection Class 1 Class 2 LDA
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F DIFS DFFS F (a) (b) F 1 L Figure
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faces of 40 subjects. • Others: A
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x DNull DRange DP (x) DNull(x) DRan
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- Page 163 and 164: Bibliography [1] FVC 2002 website.
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