EURASIP Journal on Advances in Signal Processing
Volume 2008 (2008), Article ID 492910, 16 pages
doi:10.1155/2008/492910

Copyright © 2008 Alessandro Zimmer and Lee Luan Ling. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Most of the signature verification work done in the past years focused either on offline or online approaches. In this paper, a different methodology is proposed, where the online reference data acquired through a digitizing tablet serves as the basis for the segmentation process of the corresponding scanned offline data. Local windows are built over the image through a self-adjustable learning process and are used to focus on the feature extraction step. The window's positions are determined according to the complexity of the underlying strokes based on the observation of a delta-lognormal handwritten reproduction model. Local features extraction that takes place focused on the windows formed, and it is used in conjunction with the global primitives to feed the classifier. The overall performance of the system is then measured with three different classification schemes.

1. Introduction

Online signature verification systems are extremely precise but require the presence of the author during both the acquisition of the reference data and the verification process restricting their use to specific applications [1–6]. Considering that the great majority of handwritten signatures are actually found in previously signed documents or bank checks, a different approach must be used in order to verify their authenticity. Offline systems solve these problems but lack in efficiency [3, 7–9].

In this work, a hybrid on/offline signature system was developed, where the presence of the author is required solely during the enrolment phase. After the acquisition of the online reference data, the verification process can be done directly over the desired document or bank check image. The dynamic input data serves to focus on the local feature extraction process, helping to segment the offline test data during the verification step.

The feature extraction process represents a major challenge in any signature verification system. Global features, such as the overall direction of the signature, the dimensions, and the pixel distribution, are usually not sufficient to discriminate forgeries. On the other hand, significant local features are extremely hard to pinpoint.

Great research efforts were made in order to focus on the local feature extraction process. Most of them aim at the robust extraction of basic entities called “strokes” from the original skeleton of the words [1–3, 10–12] where  the exact meaning of a stroke varies according to the segmentation method used as well as the kind of data extracted (dynamic or static).

In the proposed methodology, the definition of a stroke is made inspired by a psychomotor delta-lognormal handwritten generation model [13]. According to this model, complex neuromuscular movements such as the ones generated by the arm-forearm-hand system can be decomposed into simple impulses, also called strokes. Depending on the temporal superposition degree of such impulses, the resultant strokes can be classified according to their complexity. In this way, any sequence of points produced by ballistic-trained movements such as the ones used during the handwriting process of the signature can be consistently segmented and labeled.

The foci of attention used during the local feature extraction process of the proposed system will be located over simple stroke regions, as suggested by the handwritten generation model mentioned. Windows will be formed around those regions. The size of them will be dynamically determined during the learning process, based on an intraclass minimization criterion. The same windows will be placed over the offline data during the verification step in order to restrict the feature extraction process.

The system can be divided into the acquisition and training module (on/offline) and the verification module (offline) (Figure 1). The first one is responsible for processing the online reference data, generating the thresholds needed during the verification step. The second inputs the test image to the system, extracting similarity measurements between the reference and the test data and reaches a verdict about the authenticity of the signature.

The next session describes the segmentation methodology in detail followed by the feature extraction process and one application example with conclusion and comments.

2. Segmenting by the Stroke Complexity

Complex movements such as the ones used during the reproduction of the handwriting can be viewed as the result of a system of simple push-pull impulses produced by the neuromuscular networks involved [13]. The segmentation of the signatures will be done based on the decomposition of the individual movements forming each of the strokes producing the signature.

2.1. Modeling the Handwriting

The delta-lognormal model considered [13] sees the handwriting as a temporal superposition of individual strokes. Each of them is the result of a synchronous activation of a lognormal (asymmetric bell-shaped) impulse response system representing a synergy of agonist and antagonist neuromuscular movements that will determine, for a sequence of strokes, the speed (1), (2), and (3), and the position (4) of the writing instrument through time: (1) where equals (2) with(3)(4)where represents the amplitude of the impulse command, and the curvature and the initial direction of the movement, the starting time, the logtime delay, and the logresponse time of the neuromuscular networks involved.

Complex movements, like the handwriting, are formed by a chain of impulses more or less overlapped over the time (forming simple and complex strokes), each of them representing a synergy of push-pull reactions unleashed by the respective neuromuscular networks.

By observing (4), it can be noted that a simple stroke is characterized by a constant curvature an initial direction and a speed profile described by (2) and (3). So it is possible to separate simple from complex strokes in a consistent manner with the help of the curvature profile of the signature and its constant curvature regions, performing then the segmentation of the original dynamic signal.

The extraction of the curvature profile can be done in a number of different ways. Then, a segmentation methodology based on the speed profile obtained from the equations of the delta-lognormal model is presented. More details can be found in [14]. Later, the same profile is obtained by contour following.

2.2. Isolating Constant Curvature Regions Using the Speed Profile

The regions associated with a variable curvature are formed by the addition of simple strokes (1) and are proportional to the overlapping degree of such strokes. Those regions contain local maxima of curvature since they usually represent the transition between two strokes of different orientations or directions. Constant curvature regions are normally straight or semistraight elements (segments with a little curvature), where the pen attains maximum speed due to the lack of the restraint antagonist muscular impulse. The necessary steps needed to segment the signature are shown next.

According to the differential geometry, a continuous handwritten script can be considered as a planar curve so that the curvature can be calculated by the relation between the curvilinear and angular velocities [15]: (5)where represents the vectorial summation of the speeds of each individual stroke and the first derivative as a function of time of the direction of each stroke (angular velocity).

In the context of a specific Cartesian reference system as defined, for example, by a digitizer, the instantaneous magnitude and direction of the pen tip can be given by [16](6)(7)where (8)

To obtain the theoretical curvature value of a specific modeled continuous stroke, represented by its Cartesian coordinates, the absolute value of the instantaneous curvilinear velocity (6) as well as the value of the angular velocity must be obtained. This is achieved through the derivation of (7) as a function of time, applying it to (5). In [16], Plamondon and Guerfali adopt a simplification of this calculus for the particular case of the simultaneous activation of only two strokes. The calculus is done here for a sequence of strokes since complex handwriting movements such as the ones used to produce a signature will be analyzed. The angular velocity is then obtained by the use of (9):(9)with (10)

To exemplify the use of (9), and (10), consider the English word “she” whose interpolated skeleton is in Figure 2, sampled at a frequency of 300 Hz from a Wacom UD-1212 digitizer tablet. The word is modeled and rebuilt according to the delta-lognormal model using the parameters of Table 1, where and are equal to the amplitude of the impulse commands; and represent the initial curvature and direction of the movement; is the start time of the movement; and are the logtime delay; and and account for the logresponse time of the neuromuscular networks (the methodology of reconstruction is out of the scope of this article and can be found in [13]). By the application of (5), the curvature profile of Figure 3 is obtained. The regions of constant and variable curvature corresponding to the desired simple and complex strokes are easily distinguished.

Such segmentation methodology is efficient but could only be used if the reconstruction parameters of the delta-lognormal model for each stroke could be quickly and easily determined. It can be verified in [13] that this is a complex and time-consuming task making it unsuitable for a practical signature verification system.

On the other hand, the concept of segmentation by the stroke complexity is still valid. It can be applied if the segmentation is done directly over the curvature profile extracted from the previously acquired dynamic data. From this profile, it is possible to separate constant and variable curvature regions that correspond to simple and complex strokes according to the observations done over the delta-lognormal handwritten reconstruction model mentioned. Those regions will serve to focus on the feature extraction process in the proposed system’s architecture.

Next, the segmentation is done by contour following without the need of the parameters of the delta-lognormal model.

2.3. Segmentation by Contour Following

As mentioned in Section 2.2, the reconstruction parameters of the delta-lognormal model for a given signature are difficult to be obtained [13], so a different approach is proposed in order to segment the signature into complex and simple strokes. The contour formed through multiple recursive interpolations of the dynamic data acquired during the enrolment process of the signature is used to determine the curvature profile. The necessary steps are shown next.

2.3.1. Preprocessing

First of all, the sampled data from the digitizer is analyzed in order to separate the individual components (sections where the pen touches continuously the digitizer). Next, the one-pixel-width skeleton of the signature is constructed through multiple interpolations by splines [17] of the segmented data.

In order to reduce the digitalization noise, the pixels have to be eight-connected (i.e., the pixels should be connected to only one of its eight possible neighbors (Figure 4)). This sequence of filtered points will be used in order to calculate the desired curvature.

2.3.2. Curvature Profile

Since the curvature can be viewed as the rate of direction change of a curve, the curvature profile will be determined by the use of a differential technique known as difference of slopes (DOSs) [18]. The segmentation points (constant curvature regions) will be located within the curvature maxima obtained. The DOS method was chosen for its simplicity, and consequently rapidness in the execution of the calculations involved, and is explained as follows.

The determination of the rate of direction change is done by the displacement of a pair of segments of size secants to the trace, and is separated by a distance measuring the angular variation between them (Figure 5). The specific values of DOS parameters are application dependent and are generally determined heuristically. For the specific case of the signatures used, we found the values of the size of the component, and pixels to be appropriated.

Figure 6 shows the result of the application of DOS over the first letter of a handwritten signature (letter “c” of the alphabet). The high-frequency noise noted can be explained due to the use of the curvature calculation method over discrete angular variations. Such influence will be attenuated through the application of a discrete variable resolution convolutive Gaussian filter with the resolution  set to and a kernel of seven elements given by (11). This will reduce the intrinsic noise of the method without dislocating the constant curvature regions (Figure 7): (11)The constant curvature profile obtained will be used to segment the curvature regions into simple and complex strokes.

2.3.3. Constant Curvature Regions

The search procedure of constant curvature regions proposed is iterative, aiming at the minimization of the standard deviation around the mean value of the curvature between two maxima.

Figure 8 shows the variable regions within the curvature profile of the first noncontiguous segment of the signature (the letter “c”). It can be noted that this first segment is formed by a series of continuous strokes representing relatively constant and variable curvature regions. It is also easy to note that each constant curvature region (simple stroke) is delimited by variable curvature regions.

In those regions, according to the vectorial delta-lognormal model, there is the actuation of more than one neuromuscular impulse. This generates the temporal superposition of movements, forming, thus, the complex strokes. The algorithm developed to segment complex and simple strokes works as follows.

Initially, the mean value of all curvature points within two curvature maxima is calculated. If the difference between each extremity value of the interval and the mean is greater than a threshold, then the corresponding extreme point is eliminated, and the procedure is repeated until the standard deviation is equal or less than the desired threshold. The threshold was chosen according to the greatest possible value of (the angular variation) for a straight eight-connected segment [18] and varies with the length of every component (12). The variable was added to the original formula to compensate for the attenuation introduced by the Gaussian filter used. Its value is determined by the comparison of the mean signal intensities before and after the application of the mentioned filter: (12)Figures 9 and 10 show the complex and simple stroke regions produced by the segmentation process just presented.

This method was applied to the dynamic data of the word “she” used previously (Figure 2). The results produced by the proposed segmentation method (Figures 11 and 12 ) were compatible with those delivered by the theoretical calculation of the curvature using the equations of the delta-lognormal model (Figure 3).

In Figure 13, two signatures from the same author were overlapped and segmented into simple and complex strokes by following the aforementioned technique.

Figure 14 presents samples of the segmentation algorithm applied to different handwritten signatures. Next, the windowing process of the signature is discussed.

2.4. Windowing Process

Windows are created around the simple stroke regions segmented previously, following the general outline of the strokes and separated by a distance of from them (Figure 15). They are composed by two circle: arcs and centered in with a radius of and respectively.

If the points and belong to the same straight line, a rectangle is formed instead of the circle ring. The value of is different for each window and will be chosen and based on the minimization of the within-class error during the learning process. The length of the window is equal to the length of the correspondent simple stroke.

In order to create the windows, the skeleton and the original image have to be overlapped. The operation is done initially through a resolution change to equalize the different resolutions and by centering both in their respective center of gravity.

The center of gravity of the image is determined by first eliminating the background. This is done through a threshold operation [19] over the histogram of the gradient calculated by Sobel mask operator [20]. The remaining pixel intensity values are used as weights to center the image.

The result of those operations can be seen in Figure 16. Generally, the overlapping procedure is not perfect due to the background noise of the image and due to rounding errors, but the differences will be compensated during the learning step.

By following the stroke order, each window region is individually constructed and labeled forming an envelope around the signature that can be easily accessed during the local feature extraction process. The result of the windowing process is shown in Figure 17.

Section 3 describes the signature verification system as a whole, from the initial enrolment up to the verification of the signature.

3. System Architecture

This section contains details about the implementation of the system’s modules.

3.1. Online/Offline Hybrid Module

Six processing steps compose this module of the system. They are responsible for the acquisition and treatment of the on/offline data as follows.

Step1 (acquisition). The reference signature data is read with the help of a digitizing table in order to gather the dynamics and the image of the signature (Figure 18). The number of signatures per author acquired depends on the desired system performance (see Section 4 for detail).

Step2 (preprocessing). The input data passes through a lowpass filter in order to eliminate spurious noise inherent to the acquisition process [10]. Next, it is presegmented into components produced by pen-up/pen-down movements.

Step3 (recursive sampling). The resulting points are sampled recursively by splines [17] to generate the skeleton of the signature (Figure 19).

Step4 (segmentation into strokes). The skeleton is segmented according to the complexity of the underlying strokes [11] (Figure 20 and Section 2.3). The regions formed serve as a basis for the creation of local windows—or foci of attention—over the signal in order to pinpoint the feature extraction process.

Step5 (windowing). Windows are created around the simple stroke regions (see Section 2.4), following the general outline of the strokes (Figure 21). The length of the window is equal to the length of the correspondent simple stroke. The exact width of those windows will be determined during the learning process (Step 6 ahead).

Step6 (learning). During this step, the size of the windows is adjusted. This is done by a process aiming at reducing the within-class distance between any given pair of reference signatures acquired. The resulting signature skeleton with its personalized windows envelope will be used during the verification module of the system. Meanwhile, the mean and standard deviation of all local and global features used by the system are computed in order to be fed later to the classifier. They are as follows.

Global Features

(i) The aspect ratio (14) of the signature, calculated by the ratio of the standard deviation of the pixels in relation to the center of gravity of the signature on both the horizontal and vertical directions. (13)(14) where and represent both the horizontal and vertical dimensions of the signature, and the coordinates of each image pixel, and the center of gravity in both directions, and the total number of pixels in the image. (ii) The ratio of pixels (15) localized outside the rectangle formed by the standard deviation of the pixels previously determined and the total number of pixels in the image (15) (iii) The slant (16) of the signature defined as the maximum value of the angular frequency cumulative histogram obtained from the directions of the gradient vectors of each pixel as calculated by the convolution of the image with a Sobel mask operator [20]:(16) (iv) The distribution of the pixels around the and axes (standard deviations—(13).

Local Features.

(i) The overall direction of each stroke (18) represented by the circular mean [21] of the directional probability density function [12] calculated over each window: (17)(18) with where is the total number of segmented windows (simple strokes) of the signature. The direction of each individual signature stroke plays a fundamental role in the proposed learning process. Figure 22 shows the probability density function of the direction of the fourth window of the signature. It is easy to see that the majority of the points are biased toward 90^° (vertical direction). (ii) The ratio of pixels (19) inside each window over the total number of pixels:(19)

The distance between a given pair of reference signatures is calculated as the total difference of direction for each individual stroke of the signature. A small distance represents a great similarity between both signatures.

The reference signatures are analyzed in pairs, with the envelope containing the segmented windows of one signature overlapping the image of another. For a three-reference set, this represents a total of 6 possible combinations (Figure 23).

For any given two reference signatures and the learning process is as follows.

(1)Initially the fixed width window envelope of (generated previously) is put on top of the image of the same signature in order to calculate the mean direction of its individual strokes for a given window-size configuration, producing where is the number of windows.(2)The envelope of is put over the image of the second signature and the corresponding mean directions are calculated.(3)The total distance between both signatures is determined as (20)(4)The width of the first window is changed and Steps 1 to 3 are repeated. If the new calculated distance is smaller than the previous one, the new window size is kept in place of the older one. If not, the original width remains. Step 4 is repeated for the entire range of desired width values for each window. Experiments were done using several different ranges and the values of 80, 100, and 120 pixels worked the best for the size of signatures available. So, in this case, Step 4 is to be repeated three times for each window of the envelope.(5)The result of Step 4 above will be a customized envelope that generates the minimal distance between the two signatures and Next, another pair of envelope/image is selected, and the process repeats itself until there are no more combinations available. After the training process, there will be a set of optimized distance envelopes for all of the reference signatures. In order to allow interclass variations, the signature that presents the maximum calculated distance among the optimized envelopes will be chosen as the prototype that will represent all the signatures of the class.

After the learning process, the reference signature images can be discarded and only the envelope and the thresholds (standard deviation and mean of the extracted features) will be saved in the database.

3.2. Offline Module

The offline module will process the image of the test signature, comparing it against the reference data.

Step1 (acquisition). The test signature image is read with the help of digitizing equipment such as a scanner.

Step2 (preprocessing). The input data is filtered in order to extract it from the background using the same threshold operation described before. Depending on the application, other operations might be needed in order to eliminate horizontal lines and/or drawings from the image but they are not addressed in this work.

Step3. The correspondent window-formed skeleton is extracted from the database and is put over the image of the test signature (Figure 24), centering on the respective center of gravity of both signatures.

Step4. The extraction of the local and global features takes place, delimited by the windows of the reference skeleton.

Step5. The decision over the authenticity of the test image is taken upon the comparison between the local and global features extracted versus the thresholds stored during the learning phase.

4. Experimental Results

First of all, the discriminatory power of each feature is established with the help of a common Euclidean distance classifier (Section 4.2). Next, the system performance as a whole is tested by using the same kind of classifier arranged in a voting scheme (Section 4.4). The third experiment was done with a normalized Euclidean distance classifier. Results are obtained for three, five, and ten reference signatures, using two different data sources.

4.1. Databases

Two databases were used in order to assess the system behavior. The first one is a hybrid on/offline database made of 400 signatures from 20 authors with 20 signatures per writer. The online data (the coordinates) was acquired with the help of a digitizing tablet at a sampling rate of 100 Hz and a resolution of 1000 dpi. The subjects were instructed to sign over a white, gridlined (size-constrained) sheet of paper placed on top of the tablet in order to produce the corresponding offline data. The resulting images were scanned with a resolution of 300 dpi and at 256 gray levels. A second offline database formed by 500 additional signature images produced by other 50 volunteers with 10 samples per author was also scanned at a resolution of 300 dpi. Figure 25 shows some samples of the input data.

The signers of both databases were born in France, Canada, Brazil, Russia, Vietnam, Lebanon, Tunis, and China thus representing a broad range of signature styles.

4.2. Prototyping Individual Features

In order to analyze the discriminatory power of each individual feature, a common Euclidean distance classifier was used. Let be the set of the feature values extracted from the test signature and let be the set of the mean feature values and standard deviations obtained from the set of reference signatures during the system’s learning procedure and saved into the database. The classifier works as follows: (21)

The expression above (21) was used to test each one of the features and the correspondent with As explained in Section 3.1, can be any of the following: aspect ratio on axes, density of pixels localized outside the rectangle formed by the aspect ratio previously determined, slant of the signature, distribution of the pixels around the axes, overall direction of each stroke, density of pixels inside each window created around the simple strokes, or number of simple strokes. The proposed classifier applies a threshold of twice the standard deviation of the feature value (95% of a normal distribution) in order to label the signatures as authentic or forged.

A slightly different classifier was also implemented (22) aiming at a better evaluation of the discrimination ability of each feature. It uses the maximum and minimum feature values among all reference signatures of a given author to determine the decision threshold. The results obtained with this classifier represent the best output possible for each feature without taking into account the standard deviation of the group of reference signatures and are labeled on the figures as the ideal FAR/FRR rates.

Let be again the set of feature values extracted from the test signature . Each set of feature values obtained from every one of the reference signatures will be represented by with (number of reference signatures), extracted during the system’s learning procedure and saved into the database. The classifier works as follows: (22)

4.3. Prototyping Tests

Prototyping tests were performed using the 400 signatures from the hybrid on/offline database. False acceptance rates (FARs) and false rejection rates (FRRs) were obtained for each of the individual features by varying the acceptance/rejection threshold used.

4.3.1. Using Three Reference Signatures

The first experiment was done as follows. Three signatures from one author (from the 20 available) were chosen at random to be used by the learning algorithm described in Section 3.1, Step 6. The remaining 17 were used to help determine the FRR rate (21) and the so-called FRR ideal rate (22) explained in Section 4.2. This procedure was redone for another set of three signatures and repeated until no more signatures were available for this writer. This resulted in the average FRRs for this author. At the same time, the FAR curves for both classifiers (21) and (22) were averaged from all 20 signatures from the remaining 19 authors. The results are pictured in Figures 26(a)–26(h).

4.3.2. Using Five Reference Signatures

In the second feature prototyping experiment, five signatures from one author were used from the 20 available. As in the previously described experiment, they were chosen at random to be fed to the learning algorithm too. The remaining 15 were used to compute the FRR (21) and the FRR ideal rate (22), as explained in Section 4.2. Again, this procedure was repeated for another set of five signatures until no more signatures were available for this writer, resulting in the average FRR rates for this author. The FAR curves were also estimated using all the 20 signatures from the remaining 19 authors. The results are pictured in Figures 27(a)–27(h).

The prototyping tests just presented give an idea of the discriminatory power of each feature. The crossing points of the FAR and FRR curves point to the error rate (ERR). It can be easily observed that almost all of the individual features present an ERR of about 20% or less. The only one that presents a different behavior is the number of strokes that reached an ERR of almost 40%, showing the poorest performance of all features.

The performance of the system as a whole, unifying all the features in the same classifier, is explored in the next session using Euclidean distance classifiers.

4.4. System’s Performance

In the first experiment, multiple Euclidean distance classifiers working on a vote scheme were used to assess the performance of the system (23). If the value of a given feature was within the desired threshold calculated during the learning phase, a favorable vote was issued. The number of votes issued by all the processed features determined the acceptance or refusal of the signature. Decision thresholds of four, five, six, and seven votes were used for this classifier:

(23)

The overall performance of the system was assessed by mean error rate (MERR) curves— representing the average of FAR and FRR (24) by the equal ERRs—representing the point where the FAR equals the FRR and by the FAR/FRR curves themselves:(24)

Initially, three randomly chosen signatures from the hybrid on/offline database were used as reference with the remaining 17 signatures used as test. The results were computed, and the procedure was repeated until there were no more signatures available. Next, the test was performed using five randomly chosen signatures from the hybrid on/offline database as reference with the remaining 15 signatures used as test. Again, the procedure was repeated until there were no more signatures available.

The mean error rate curves presented in Figures 28 and 29 are the average of the values obtained for three and five reference signatures, respectively, using the vote-based classifier just mentioned considering decision thresholds of four, five, and six votes. Figures 30 and 31 show the corresponding FAR/FRR curves.

Using the same vote-based classifier but taking into account all the eight features, the system produced equal error rates of 9.31% for a threshold of 4 votes, 8.86% for 5 votes, 10.31% for 6 votes, and 10.89 considering 7 votes. With five signatures as references (15 as test), the results improved (except for the case of 4 votes) with ERR of 7.21% for 4 votes, 5.89% for 5 votes, 5.53% for 6 votes, and 6.07% considering 7 votes.

In the next experiment, the 500 unseen signatures from the second image-only database were used as random forgeries instead of the remaining signatures of the first database, working with the same vote-based classifier. The system produced equal error rates of 8.19% for 4 votes, 7.77% for 5 votes, and 8.33% using 6 votes. With five signatures as references (15 as test), the results improved with ERR of 6.21% for 4 votes, 4.20% for 5 votes, and 4.67% for 6 votes (Table 2).

Next, a normalized Euclidean distance classifier (25) was used where the decision was taken based upon the sum of all the distances calculated among each one of the individual features. Tests were done with the hybrid database considering three, five, and ten reference signatures producing equal error rates of 10.84%, 5.41%, and 1.42%, respectively. Figure 32 shows the corresponding FAR/FRR curves: (25)

5. Concluding Remarks

A hybrid signature verification merging online and offline data was implemented. The main advantage of this approach is the ability to use the dynamic data in order to robustly segment the signature into windows that will be used later during the verification process. This will allow these financial institutions such as banks to acquire online data through the usual enrolment process in a supervised manner and to use this information later to recognize the client’s signature in checks and other documents without the presence of the bearer of the signature. The segmentation is not random but is based on the premises derived from a handwritten reproduction model.

The proposed system used two databases totaling 900 signatures from signers of eight different nationalities. It was able to correctly segment the signatures producing equal error rates of about 5% for 5 reference signatures and about 1% for 10 reference signatures by the use of common Euclidean distance classifiers. Further improvement on the results could be achieved by the analysis of the regions around complex strokes as well as by the use of a more efficient classifier.

Acknowledgments

The authors would like to thank professors R. Sabourin (Laboratoire LIVIA/École de Technologie Supérieure de Montréal) and R. Plamondon (Laboratoire SCRIBENS/École Polytechnique de Montréal) for their useful guidance and remarks during the development of this work. It is also a pleasure to acknowledge the financial support from the Brazilian government (CAPES/CNPQ).

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