Hand-written Signatures by Conic s Representation

Transcription

1 Hand-written Signatures by Conic s Representation LAUDELINO CORDEIRO BASTOS 1 FLÁVIO BORTOLOZZI ROBERT SABOURIN 3 CELSO A A KAESTNER 4 1, e 4 PUCP-PR Pontifícia Universidade Católica do Paraná PUC-PR - Rua Imaculada Conceição, 1155, Curitiba, Paraná, Brasil : CEP : bastos@rla01pucprbr fborto@rla01pucprbr 4 kaestner@rla01pucprbr 3 Université du Quebéc, École de Technologie Supérieure, Département de Génie de la Prodution Automatisée, 4750 Henri-Julien, Montréal QC, Canadá, HT C8 sabourin@gpaetsmtlca Abstract This paper presents a representation of hand-written signatures by conic s, like straight lines, ellipses and hyperbole s This representation allows a simplification of drawn of the signature, for the purpose of verification in the context of random forgeries, when forger doesn't imitate the original signature 1 Introduction During the last two decades it there has been a lot of research in the field of manuscript signatures The major research in this field has been done in signature verification systems that from an original signature of a person, try to identify if a signature analysed is true or false A lot of security and financial reasons justify the research in this field, like the verification of checks, transactions with credit cards and public documents [SABO90], [PLAM90], [PLAM89], [BRAU93], [RAND90] Besides, signature verification is considered one of the best ways that a automatic personal identification system can be based, because the signature must be "produced" by a person, on the contrary of passwords and identification cards that are simply "processed" and can be lost or stolen The main purpose of this work is contribute to developing a real time system for signature verification, by means of a new representation of hand-written signatures by conic s, like straight lines, ellipses and hyperbole s This method permits a simplification of signature tracing, for the purpose of verification in the context of random forgeries, when forger doesn't imitate the original signature Pre-processing The signatures utilised here have 56 grey levels and are 51 pixels wide and 18 pixels high For instance, we utilise the original signature shown in figure 1 for extraction of its equations, where an equation represents a part of signature tracing First, we apply the morphological process of Tophat [FACO93], to increase the contrast between signature tracing and background (figure ) After this, we utilise the thresholding method of Otsu [OTSU79], finding the result shown in figure 3 Finally, the Zang and Suen s thinning process [GONZ87] provides the final result of pre-processing (figure 4) Figure 1 Original signature Figure Original signature after Tophat process

2 After finding the junction points and end points, we can extract the points of a signature tracing by means of the Freeman algorithm [GONZ87] with eight directions We start with a junction or end point, following the signature tracing up to find another junction or end point For instance, we take the signature in figure 41 Figure 3 Signature of figure after the thresholding method of Otsu (x1,y1) Area utilized to modeling the tracing Figure 4 Signature of figure 3 after thinning process 3 Extraction of Characteristic Points To exemplify the extraction of equations of a signature, we need some definitions: a) Transition Function: each pixel P i of a skeleton, resulting from the thinning process, has a transition function T(P i ) associated to it T(P i ) represents the connectivity between P i e its eight neighbours and T(P i ) is defined like the number of transitions of 0 (white) to 1 (black) when the eight neighbours of P i (that is, P 1, P,, e P 8 ) are traced in a clockwise direction b) End Points (EP): an end point (EP) is a pixel P i with T(P i ) = 1 c) Junction Points (JP): a junction point (JP) is a pixel P i with T(P i ) 3 After the thinning process, the signature tracing is ready to that we find the junction points and end points Then, by means of an inspection method, the junction points and end points are found (figure 31) End Points Junction Points Figure 31 Junction points and end points (xn,yn) Junction and end points Signature tracing Figure 41 Area to extraction of a mathematical equation The point with co-ordinates (x 1,y 1 ) is an end point and the point with co-ordinates (x n,y n ) is a junction point Starting with the end point and following the signature tracing up to the junction point, we obtain all necessary points for modelling the mathematical equation After finding the tracing points, the minimum square method of curve adjusting is applied over all points between two characteristic points The mathematical modelation of this curve is made by conic s general equation [BOUL86]: G( x, y) = A' x + B' xy + C' y + D' x + E' y + F' = 0 Dividing the equation above by A' 0, to obtain an independent term, we have: x + Bxy + Cy + Dx + Ey + F = 0 4 Extraction of Mathematical Equations

3 Bxy + Cy + Dx + Ey + F = x We know the set of tracing points x and y, then we can find the values of B, C, D, E and F Then, if we know 5 points or more, we have a system with 5 variables and the number of equations is equal to the number of tracing points This system can be represented by: xy 1 1 y1 x y xy y x y x y y x y x B 1 x C D = E F 1 x 1 n n n n n n or AX = Y We can solve the system above by means of the last square method of curve adjusting [NOBL86]: X= ( A T 1 T A) A Y Conic s can be classified in three types [BOUL86] shown in table 41 Type Condition Species Ellipses B' 4A' C' < 0 empty, point, circle or ellipse Parabola B' 4A' C' = 0 straight line, reunion of two parallel straight lines or parabola Hyperbole B' 4A' C' > 0 reunion of two crossing straight lines or hyperbole Table 41 Classification of conic s Taking again the signature of figure 4, after finding all the junction points and end points, we apply the minimum square method of curve adjusting to each set of tracing points After this, we obtain 4 equations Figure 4 illustrates the location of equations in the signature Figure 43 shows the 4 equations found by the minimum square method of curve adjust Figure 4 Location of equations in signature Nº Equation Type 1 x -974xy+7y x+3043y-99368=0 H x +857xy+188y x-0088y+46663=0 E 3 x +1035xy+674y x-44996y+17419=0 H 4 x -063xy+0966y +687x-17956y =0 H 5 x -01xy+3801y -8841x-4377y+86136=0 E 6 x -0339y+0080y -14x y+5143=0 E 7 x x y+19996=0 P 8 x +190xy+1189y -348x-43y+141=0 P 9 x -0318xy+004y -863x y+18559=0 P 10 x -100xy-40001x+4000y =0 H 11 x +139xy+0540y x-7434y+594=0 E 1 x -0616xy+04334y -478x-10y+13466=0 E 13 x -5xy+133y +307x-1891y =0 H 14 x +3945xy+33497y -91x-589y+091=0 H 15 x +0789xy+13y -871x-10184y+0083=0 E 16 x +614xy+9474y -419x-1693y+4387=0 H 17 x -1977xy-1691y +35x+49684y-69103=0 H 18 x -300xy+1833y x+31665y-16665=0 H 19 x -104xy+64y -956x-1387y+778=0 E 0 x -3104xy+808y x-8760y+641=0 E 1 x xy+1y -11x-11y+30=0 P x +0307xy+049y x-1791y+34641=0 E 3 x -043xy+1003y -043x+169y+105=0 E 4 x -0489xy+04339y -917x-531y+6313=0 E Figure 43 Equations found in the signature of figure 4 H is hyperbole, P is parabola and E is ellipse Figure 44 illustrate the original tracings of signature compared with the equations obtained Figure 45 presents a comparison between the signature tracing after the thinning process and the reunion of all equations found by the minimum square method of

4 curve adjusting Figure 46 presents a superposition of the two signatures of figure 45 Making a superposition between the reunion of all equations found and the signature after the thresholding method of Otsu, among the 878 points found by the equations, 761 are in common with the signature after thresholding method This results in a similarity index of 86,7% Number Original tracing Corresponding equation Number Original tracing Corresponding equation Figure 44 (continuation) (a) 1 13 (b) Figure 45 Comparison among signatures (a) Signature tracing after the thinning process (b) Reunion of all equations found by the minimum square method of curve adjusting Figure 44 Original tracings of signature compared with the equations obtained

5 Figure 46 Superposition of the two signatures of figure 45 5 Testing the Modelation Method To examine the modelation method presented here, 0 different signatures of 6 different people were utilised, totalling 10 signatures We made a superposition between the reunion of equations found and the respective signatures after the thresholding method of Otsu For each person a mathematical mean was determined among the results obtained with the 0 signatures The results found are presented in table 51 Person Similarity index 1 87,8% 97,3% 3 89,3% 4 88,6% 5 86,3% 6 9,4% Table 51 Observation: The signature of figure 1 belongs to person 1 6 Conclusions The minimum square method of curve adjusting presents good results in mathematical modelation of hand-written signatures This can be proved by the similarity indexes presented in table 51 However, when the number of points of a tracing is small (5 to 10 points), there is a variation among the tracings and the equations As proposals for new projects we suggest: a) the development of fuzzy grammars based in the most significant tracings of signatures and in the relative position between them, utilising the approach presented here; b) the process presented here can be used in other areas of pattern recognition, like medical images or computer vision [BOUL86] Paulo Boulos, Ivan de Camargo e Oliveira, "Geometria Analítica: um Tratamento Vetorial", McGraw-Hill, São Paulo, 1986 [BRAU93] Jean-Jules Brault, Réjean Plamondon, "A Complexity Measure of Handwritten Curves: Modeling of Dynamic Signature Forgery", IEEE Transactions on Systems, Man and Cybernetics, vol 3, nº, 1993 [FACO93] Jacques Facon, "Processamento e Análise de Imagens", VI Escola Brasileiro-Argentina de Informática [GONZ87] Rafael C Gonzalez, Paul Wintz, "Digital Image Processing", Addison-Wesley Publishing Company, 1987 [NOBL86] Ben Noble, James W Daniel, "Álgebra Linear Aplicada", Editora Prentice/Hall do Brasil, Rio de Janeiro, 1986 [OTSU79] Nobuyuki Otsu,"A Threshold Selection Method from Gray-Level Histograms", IEEE Transactions on Systems, Man and Cybernetics, vol SMC 9, nº 1, pags 6 a 66, 1979 [PLAM89] Réjean Plamondon, Guy Lorette, "Automatic Signature Verification and Writer Identification - the State of the Art", Pattern Recognition, vol, nº, pag 107 a 131, 1989 [PLAM90] Réjean Plamondon, Guy Lorette, Robert Sabourin, "Automatic Processing of Signature Images: Static Techniques and Methods", Handwritten Pattern Recognition, 1990 [RAND90] David Randolph, Ganapathy Krishnan, "Off- Line Machine Recognition of Forgeries", Machine Vision Systems Integration in Industry, vol 1386, pags 55 a 64, 1990 [SABO90] Robert Sabourin, Réjean Plamondon, Guy Lorette, "Off-Line Identification with Handwritten Signature Images: Survey and Perspectives", SSPR, References

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