Handwriting Trajectory Movements Controlled by a Bêta-Elliptic Model

Handwriting Trajectory Movements Controlled by a Bêta-Elliptic Model Hala BEZINE, Adel M. ALIMI, and Nabil DERBEL Research Group on Intelligent Machines (REGIM) Laboratory of Intelligent Control and Optimization of Complex Systems National School of Engineers of Sfax, BP. W, 3038, Sfax, Tunisia Phone: +6-74-74.088, Fax: +6-74-75.595 Emails: {Hala.Bezine, Adel.Alimi, N.Derbel}@ieee.org Abstract This paper studies the dependence between geometry and kinematics in handwriting generation Movements. Based on a handwriting generation model developed earlier, movements parameters are extracted directly from on-line recording of velocity profiles. These parameters, which reflect dynamic process, are found to be highly related to the geometry of handwriting movements. The overall approach is based upon the hypothesis that complex human movements can be segmented into, basic and simple units. In other words, due to the intrinsic properties of the neuromuscular system involved in a rapid writing task, there is a class of simple movements, hereafter called strokes. More complex movements are thus generated by the addition of the various strokes belonging to such a class. Keywords: Cursive Handwriting, Handwriting Modeling, Bêta-Elliptic Model.. Introduction To date, relatively few studies have been carried out to explore systematically the relations between handwriting velocity, coordination of writing movements: rhythm and the quality (accuracy and deformations) of the writing product [3, 4]. In this paper, first of all, we propose a handwriting model, which is based upon several assumptions. First, it supposes that fast handwriting, like any other highly skilled motor process, is partially programmed in advance. Consequently, it is easy to understand that a human subject can anticipate what will happen once his neuromuscular system has been fed by a command. Second, it assumes that movements are represented and planned in the velocity domain, since the most widely accepted invariant in movement generation is the Beta shape of the velocity profiles. In consequence, in the current paper, we are concerned with the following problems: To investigate the relation between the instantaneous tangential velocity and the form of trajectory generated by a handwriting movement. To study the relation between kinematics and form for the case of handwriting movements taking into account the overlapping of the beta velocity profiles, for this case elliptic trajectory are designed for this purpose. The paper is organized as follows: section summarizes the Bêta model adopted to describe the kinematics of the cursive handwriting. Section 3 illustrates the relationship between kinematics and form for the case of handwriting movements and elliptic trajectories are designed for this purpose. Eventually, conclusions are given in section 4.. Modeling Velocity Profiles The main goal of the experiments reported in this article is to further the investigation of the constraints between trajectory and kinematics, which provide a clue to both the degrees of freedom problem and the computational complexity problem [0]. In addition, the model proposed here is based upon several Proceedings of the Seventh International Conference on Document Analysis and Recognition (ICDAR 003) 0-7695-960-/03 $7.00 003 IEEE

assumptions. First, it supposes that fast handwriting, like any other highly skilled motor process, is partially programmed in advance. Second, it assumes that movements are represented and planned in the velocity domain, since the most widely accepted invariant in movement generation is the Beta shape of the velocity profiles. The serial nature characterizing the central representation of a movement is also detectable from a kinematics point of view during the generation of the movement in the external environment, where the velocity profile of the end effector is decomposable into a series of basic, approximately Beta, asymmetric profiles overlapping each other as a function of the speed of the whole movement [ 6]. Based on the Delta-Lognormal theory of Plamondon [7], in which a neuromuscular synergy is composed of two parallel and global systems which represent, respectively, the set of neural and muscular networks involved in the generation of the agonist and antagonist activities resulting in a specific movement. In fact, supposing that each of these two systems is composed internally of n neuromuscular subsystems characterized by an impulse response that is real, normalized, and non-negative. If n is sufficiently large, applying the central limit theorem, the global impulse response will converge to a lognormal curve, and finally, the complete velocity profile of the neuromuscular synergy will be described by a Delta-Lognormal law [, ]. In this context, for our case, the generation of a complex trajectory pattern is the result of the activation of n neuromuscular subsystems characterized by an impulse response that is real, normalized, and non-negative. If n is sufficiently large, applying the central limit theorem, the global impulse response will converge to a Beta curve. Consequently the complete velocity profile of the neuromuscular system will be described by a Beta law (see Equation ). n V σ ( t ) = β i ( t ) () i = with: p q t t t tc t t t p q β (, 0,,, ) = t t c tc t () 0 and p * t + q t c = p + q 0 * t (3) where t 0, t, t c represent, respectively, the initial time, the final time and the time where the velocity reaches its maximum. Finally, to test the validity of this model, several experiments were performed, according to the following concept: The database of words captured for our study comes from a Wacom 4 electronic digitizing tablet. The raw data x(t) and y(t) collected from the digitizer was sampled at 00 Hz. These signals were filtered with a Chebychev second-order low-pass filter with a cut-off frequency of about Hz. The curvilinear velocity V(t) (see Equation 4) is then computed using a second-order derivative filter with finite impulse response (FIR). dx( t) dy( t) V ( t) = + (4) dt dt Handwritten characters are then segmented into simple movements as already mentioned, called strokes, and are the results of a superimposition of time-overlapped velocity profiles. The curvilinear velocity of each individual stroke obeys the Beta approach. So, the generation of a complex trajectory pattern is the result of an algebric addition of stroke velocity terms (see Equation 5). v(t) vi ( t0i) = n i = t (5) However, the difficulty of analyzing natural handwriting comes mainly from retrieving the strokes and is due to such reasons: firstly, for the case of rapid handwriting movement, the majority of strokes are partially hidden in the measured signal, as a result of the time-overlapping process and, secondly, there exists an infinite number of combinations of strokes that can be investigated to generate the same complex movement [7, 8]. To extract the set of parameters that best fits the handwriting velocity signals, we need follow several Proceedings of the Seventh International Conference on Document Analysis and Recognition (ICDAR 003) 0-7695-960-/03 $7.00 003 IEEE

steps according to the Beta model. First, in order to estimate the number of strokes and their approximate location, an analysis of the extrema and inflexion points of the curvilinear velocity signal is applied. Second, we combine a heuristic method and a graphical approach to locally estimate the 5 parameters of each stroke. Third, once the initial local parameters have been estimated, a simultaneous global optimization of all parameters (5 parameters per stroke) is carried out, taking the time overlapping effect into account. For this purpose, the Least-square non-linear optimization technique is used. The system parameters was extracted, minimizing the least square errors between the original curvilinear velocity and the model predictions. Where the considered function to optimize is F such that: n F = V ( t) β i = i (6) Figure shows an example of the word elle as written by a human subject and as generated by the algebric Beta model using the parameters extracted by the method explained above. Figure depicts the curvilinear velocity profile: the dotted curves superimposed on this graph represent the regenerated output from the parameters extracted with the model from the original velocity curve 6 4 0 8 6 4 Figure : Example of the Word «elle». Amplitude (cm/sec) 0 0 0.5.5.5 Time(sec) Figure : Curvilinear Velocity Profile Solid Line : Original Curvilinear Velocity Dushed Line : Reconstructed with Beta Shapes 3. An Explanation for the Feature of a Handwriting Trajectory Movement Controlled by a Bêta- Elliptic Model Proceedings of the Seventh International Conference on Document Analysis and Recognition (ICDAR 003) 0-7695-960-/03 $7.00 003 IEEE

As already mentioned in the previous section, the sequence of motor strokes can be described by an algebraic sum of velocity components, where a Beta approach describes each one. These results in a correspondence between the minimum of the curvilinear velocity and the maximum of curvature as reported by many (see, for example, [5], [6] and [9]). The problem now is the determination of the curvature function of the script contour. In the literature, a few models have already been proposed in this direction: in [], for example, the cursive handwriting is considered as the concatenation of circle segments characterized by their center and rayon defining each one. Analyzing the curvature function, the curvilinear velocity varies with handwriting cycle: curvilinear velocity decreases for the parts of minus curvature of handwriting script and increases for the parts of plus curvature of handwriting script. For this reason, we proposed more general structure of the contour curvature: the elliptic form verifying the equation 7, where X(t) and Y(t) are the Cartesian coordinates and a and b are the huge and small axes respectively. Y X a + Y b y = X (7) Theoretically, to design an elliptic segment, representing a stroke in the velocity domain, we must have at least 5 points. We are based on the dynamic aspect of the handwriting script to determine the 5 points of each stroke. In fact, the extreme points of an elliptic segment correspond in temporal space to t 0 and t c of the considered Beta profile. The lasting three points are constrained to belong to the same considered Beta profile, in other words the same stroke. Having obtained the 5 points, the problem now is to check the different parameters of the considered elliptic segment (x 0, y 0, a, b,θ ), that are determined numerically. An example of generation of simple elliptic arc is presented in figure 4, where the 5 points are deduced from the curvilinear velocity profile of figure 5 reconstructed with Beta shapes (dashed line). The original handwriting script corresponds to the continuous line of figure 4. Thus a handwriting script may be considered as the superimposition of strokes in time. A curvilinear velocity that obeys the Beta approach characterizes each stroke. Then an elliptic arc is overlapped in time with the previous one and a stroke is thus an elliptic arc determined by nine parameters: x 0, y 0, a, b,θ, t 0, t, p and q. In fact these strokes are not directly apparent in the image of a handwritten script. There are partially hidden in the trajectory, as a sequence of the superimposition process. b a θ x O Figure 3: Descriptive Scheme of the elliptic form. Figure 4: Approximation of Handwriting Script with Elliptic Contour... Handwriting Script Elliptic Contour Proceedings of the Seventh International Conference on Document Analysis and Recognition (ICDAR 003) 0-7695-960-/03 $7.00 003 IEEE

References [] Alimi A. Contribution Au Développement d Une Théorie de Génération de Mouvements Simples et Rapides Application Au Manuscrit, Ph. D. Dissertation, Ecole Polytechnique de Montréal, 995. [] Guerfali W. Modèle Delta-Lognormal Vectoriel pour l Analyse du Mouvement Et La Génération de l Ecriture Manuscrite, Ph. D. Dissertation, Ecole Polytechnique de Montréal, 996. [3] Grossberg, S. and Paine R. W. A Neural Model of Cortico-Cerebellar Interactions During Attentive Imitation and Predictive Learning of Sequential Handwriting Movements, Neural Networks, vol. 3, pp. 999-046, 000. Figure 5: Approximation of curvilinear velocity with Bêta shape, case of simple arc. original velocity.. reconstructed one with Bêta shape. 4. Discussions and Conclusions In this paper we have presented an approach to model cursive handwriting, which exploits the two properties kinematics and geometry. Based on the fact that kinematics and form are entirely dependent, we have suggested that a handwriting trajectory can be described as the concatenation of simple strokes. Combining dynamic and form: a stroke is an elliptic arc, approximated by a curvilinear Beta velocity profile and determined by nine parameters: x 0, y 0, a, b,θ, t 0, t, p and q. The experiments have shown that the Beta model is able to reproduce the kinematics properties of handwriting script. So a handwriting script is considered as the concatenation of elliptic strokes where each one overlapped its former. At last, but not least, it is very important to notice that the contour approximation rate with elliptic form exhibited by the system differ from case to another of handwriting and mainly from stroke to another belonging to the same handwriting script and this is due to the choice of the 5 points and precisely the three internal points. In fact the three intermediate points determine the curvature of the stroke. [4] Hollerbach, J. M. An Oscillation Theory of Handwriting, Biological Cybernetics, no. 39, pp. 39-56, 98. [5] Plamondon, R. Stelmarch Yu L. D. and Clement, B. On the Automatic Extraction of Biomechanical Information from Handwriting, Signals IEEE Trans. On Systems, Man, and Cybernetics, vol., pp. 90-0, 99. [6] Plamondon, R. The Generation of Rapid Human Movements. Part II: Quadratic and power Laws, Technical Report EMPR/93-93/5, Ecole Polytechnique de Montréal, 993. [7] Plamondon, R. A Kinematics Theory of Rapid Human Movements. Part I: Movement Representation and Generation, Biological Cybernetics, vol. 7, pp. 95-307, 995. [8] Plamondon, R. and Guerfali W. The Generation of Handwriting with Delta-Lognormal Synergies, Biological Cybernetics, vol. 78, pp. 9-3, 998. [9] Viviani, P., and Schneider, R. A Developmental Study of the Relation between Geometry and Kinematics in Drawing Movements. Journal of Experimental Psychology: Human Perception and Performance, vol. 7, n., pp. 98-8, 99. [0] Wada, Y., and Kawato M. Quantitative Examinations for multi Joint arm Trajectory Planning- Using a Robust Calculation Algorithm of the Minimum Commanded Torque Change Trajectory. Neural Networks, vol. 4, pp. 38-393, 00. Proceedings of the Seventh International Conference on Document Analysis and Recognition (ICDAR 003) 0-7695-960-/03 $7.00 003 IEEE