Thesis - Instituto de Telecomunicações

4.1. HUMAN-COMPUTER INTERACTION 63In order to smooth the signal in the spatial domain, we applied the three following stepsin sequence:• a linear space interpolation that produces a uniformly spaced vector ( px ˆ i , py ˆ i ), witha spacing interval γ =1pixel. The size of this vector is ˆn = s n−1γ ;• a cubic spline smoothing interpolation that results in the curve ( px ˜ i , py). ˜ The beginningand ending points of the stroke are fixed points in this interpolation, (px 1 , py 1 )and (px n , py n ), respectively. The length of the smoothed curve is denoted by ˜s.• A final uniform spatial re-sampling of the smoothed curve ( px ˜ i , py ˜ i ), leading to thecurve points (px ∗ i , py∗ i ). Associated with this vector is the path length vector, s∗ i ,i =1, . . . , n ′ , n ∗ = ˜s γ , computed from points (x∗ i ,y∗ i ) as in equation 4.1.Figure 4.2 illustrates the preprocessing phase. In this figure, black dots (•) represent theoriginal sampled points, open dots (◦) correspond to the linearly interpolated points. Thecontinuous line is the resulting smoothed data.Spatial Information We define six vectors in the spatial domain over the smoothed,uniformly sampled curve points (x ∗ i ,y∗ i ):• px ∗ — horizontal coordinates.• py ∗ — vertical coordinates.• s ∗ — path distance from the origin.• θ — angle of the path tangent with the x-axis.• c — curvature• c ′ — curvature derivative with respect to space.The ′ (prime) notation identifies first order derivative and ′′ (double-prime), identifiesthe second order derivative or the first and second order difference equations in the case oftime discrete signals.The first three vectors were obtained during the smoothing process. The simplest formulato compute the angle of two consecutive points is not continuous near π and −π.