A fingerprint verification system based on triangular matching and ...

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KOV ACS-VAJNA: A FINGERPRINT VERIFICATION SYSTEM BASED ON TRIANGULAR MATCHING AND DYNAMIC TIME WARPING 1271Fig. 7. Original image on the left, distorted image on the right with local deformation less than 10 percent, but consequent global deformationreaching 45 percent.45 percent extension of edge GH. This simple (andexaggerated) example shows the idea of the model, whichdefines the distortion of <strong>fingerprint</strong> images where a smalllocal distortion is amplified on a global basis.The following algorithm (written in a pseudo C code) isused to extract a possible minutiae set where the matching isestablished. The result is a number of matching minutiaepairs on the reference and test images, which according tothe required number of pairs, forms the basis of theidentification. By generating a map of already consideredminutiae, it is possible to significantly reduce the number oftentative matching maps and employ a very short computationtime in this phase.The following list contains the functions and proceduresused in the algorithm:. GetNextMinutiae() gets the next, already notconsidered, reference minutiae from the ordered setdefined in Section 2.1.3.. GetNextTestMinutiae(ref) gets the next, alreadynot considered, test minutiae corresponding toref minutiae from the ordered set with decreasingsimilarity between test and reference regions accordingto Section 2.2.1.. LineLength(ref1, ref2, test1, test2) computesthe length difference between the linesconnecting two minutiae in the reference image andtwo in the test image. This procedure may includethe <strong>verification</strong> phase shown in the next section toimprove performance.. ComputeAngle(ref1, ref2, test1, test2)computes the angle between two lines connectingminutiae pair ref1, ref2 and pair test1, test2,respectively.. AddMatchedMinutiae(ref1, test1, ref2,test2) adds the pairs of matched minutiae inreference image and in test image to the alreadymatched minutiae list.. GetNextMinutiaePair(ref1, test1, ref2,test2) gets the next minutiae pair belonging to analready matched line both in reference and testimages.. GetNextCloseMinutiae(ref1, ref2) gets thenext,already not considered reference minutiae fromthe ordered minutiae list minimizing the functionmax(L2(ref1, ref3), L2(ref2, ref3)), wheremax() is the larger between two real numbers andL2() is the Euclidean norm on the bidimensionalCartesian plane.. ComputeTestCoordinates(X, Y, ref3, ref1,test1, ref2, test2) computes the …X; Y † coordinatescorresponding to ref3 using the linearroto-translation defined by the mapping ref1 >test1, ref2 > test2 according to the formulae:whereX ˆ x 3 ‡ y 3 ‡ x Y ˆx 3 ‡ y 3 ‡ y ; ˆ …X 2 X 1 †…x 2 x 1 †‡…Y 2 Y 1 †…y 2 y 1 †…x 2 x 1 † 2 ‡…y 2 y 1 † 2 ˆ …X 2 X 1 †…y 2 y 1 †…Y 2 Y 1 †…x 2 x 1 †…x 2 x 1 † 2 ‡…y 2 y 1 † 2 x ˆ X 1 x 1 y 1 y ˆ Y 1 y 1 ‡ x 1with the uppercase variables referring to the testcoordinate space and the lowercase coordinates tothe reference coordinate space.. OrderTestMinutiae(ref, x, y) orders the testminutiae corresponding to the reference minutiae refaccording to the increasing distance from point ofcoordinate (x, y) in the test image.. NumberOfMatchedMinutiae() computes thenumber of already matched minutiae.. MaxDistance is the maximum allowable deformationbetween segments (for example, 10 percent or6 pixels).. MaxAngle allows to control the maximum rotation(for example, 20 degrees).. MinimumMatch is the minimum number of requiredminutiae to match (for example, 14) for a valididentification.
1272 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 22, NO. 11, NOVEMBER 2000Fig. 8. Matching triangle-<strong>based</strong> graphs: reference image on the left, test image on the right./* Triangular Matching Algorithm*/:do/* sets two minutiae on reference image */ref1 =GetNextMinutiae();ref2= GetNextMinutiae();/* sets the corresponding two minutiaeon test image */dotest1 = GetNextTestMinutiae(ref1);dotest2 = GetNextTestMinutiae(ref2);if (test2 != null)dist = LineLength(ref1, ref2, test1,test2);angle = ComputeAngle(ref1, ref2, test1,test2);endif/* the correspondence is valid ifdeformation issmall and rotation is allowed */until (dist < MaxDistance ANDangle < MaxAngle)OR test2 == nulluntil test2 != null OR test1 == nullif (test1 == null OR test2 == null)goto end_NO_MATCH; /* no matching can bedefined */endif/* adds minutiae pair to alreadymatched list */AddMatchedMinutiae(ref1, test1, ref2,test2);/* sets third minutiae for each pairof reference andtest matching minutiae pairs according toroto-translation defined bythe minutiae pairs */dovalidpair=GetNextMinutiaePair(&ref1,&test1,&ref2, &test2);if (validpair)doref3 = GetNextCloseMinutiae(ref1, ref2);if (ref3 != null)/* defines third minutiae correspondingarea on test image */ComputeTestCoordinates(&x,&y, ref3,ref1,test1,ref2, test2);OrderTestMinutiae(ref3, x, y);endifdotest3 = GetNextTestMinutiae(ref3);if (test3 != null)dist1 = LineLength(ref1, ref3,test1, test3);dist2 = LineLength(ref2, ref3,test2, test3);angle1 = ComputeAngle(ref1, ref3,test1, test3);angle2 = ComputeAngle(ref2, ref3,test2, test3);endif/* the correspondence is validif deformation issmall and rotation is allowed */until (dist1 < MaxDistance ANDdist2 < MaxDistance ANDangle1 < MaxAngle ANDangle2 < MaxAngle) ORtest3 == nulluntil test3 != null OR ref3 == nullif (ref3 != null AND test3 != null)AddMatchedMinutiae(ref1, test1,ref3, test3);AddMatchedMinutiae(ref2, test2,ref3, test3);endifendifuntil validpair == falseuntil NumberOfMatchedMinutiae() >MinimumMatch/* continue with validation procedure of nextsection */
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A fingerprint verification system based on triangular matching and ...

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