3.6 Treadmill experiments
3.6.6 Biometric identification
Researchers have previously used magnitude weighted phase as a suitable feature vec- tor [18, 17, 19, 117, 119] for biometric identification. Their results suggest that magni- tude weighted phase achieves better discrimination between subjects, when compared to components of magnitude or phase alone. Lower order harmonics seem to be more significant, encode the gross features of cyclic motion and are less susceptible to ex- perimental noise than those of higher orders. Random measurement noise and natural fluctuations in day to day subject motion patterns account for the majority of these differences within the higher order harmonics.
Values of phase are normalized modulo 2πto the range (−π :π), hence phases distributed around both extremes of the range must be treated with caution. Naively matching coefficients of magnitude weighted phase in order to facilitate recognition is asking for trouble. Each phase direction vector corresponding to each of the harmonic contributions is of unit length, hence all vectors carry an equal weighting. Two putatively similar phase vectors p and q can be matched by computing a cost error C based on the dot product between both direction vectors: C= 12(1−p⊤
q). This cost error lies within the range (0 : 1). Since all phase vectors have equal weighting, then the sum of all residual costs, between the set of phases in two gait feature vectors, is susceptible to a level of noise contamination from the higher order harmonics. We can assign a probabilistic weighting factor of significance ωk to each of the harmonics. The obvious choice is to
assign the significance weights from the tail of some suitable Gaussian (an exponential function). However, should we use the same Gaussian for each person? What happens if the amplitude of a phasor happens to be zero?
The first thing we notice about the distribution of the set of normalized amplitude coef- ficientsbk is that they are log-linear, illustrated by the straight line plots in figure 3.33,
hence are of the required exponential form. Since we have also normalized the set of coef- ficients so that the first componentb1is unity, then the normalized amplitude coefficients
bk can be used directly as the required set of significance weights ωk. These weighting
factors are perforce invariant to changes in gait speed, stride length and cadence, thus are ideal for comparative purposes over the range of required walking conditions. We have identified a suitable number of invariant features that can be used for subject identification. The static parameters of articulated leg motion include the normalized limb segment lengths (d2,· · ·), and corresponding normalized amplitude and phase coef- ficients of the modified Fourier series leg functions (b2, ψ2,· · ·). We can then compute a Euclidean distance metricρ=kPb −Pk, between the measured biometric feature vector Pof subject motion and a feature vector Pb stored in the database.
ρ2 =λ1 m X j=2 ³ b dj−dj ´2 +λ2 n X k=2 ³ bbk−bk ´2 +λ3 n X k=2 µ bk 2(1−bv ⊤ kvk) ¶2 + · · · (3.23)
where the phase direction vectorsbvand vare commuted from polar to Euclidean form, m is the number of segments in the articulated limb model, and n is the number of Fourier harmonics used to represent the limb motion. We can also pick the respective weighting factors λi to bias the fitting error in favour of any particular set of feature
coefficients. b vk = (cosψbk,sinψbk) ⊤ (3.24) vk = (cosψk,sinψk) ⊤ (3.25) N X i=1 λi = 1 (3.26)
Note that the order of magnitude of each set of coefficients is similar, since all biometric parameters have been normalized so that the set of first coefficients are unity: d1 = 1,
b1 = 1 andvb1·v1 = 1 (first phase angles are zero).
Our results are similar to the works of BenAbdelkader and Tanawongsuwan in the fact that the set of fundamental amplitude termsa1is proportionally similar to stride length, and fundamental frequency f0 is proportional to cadence. We choose to unit normalize these parameters in order to retain a consistent scaling over different walking speeds. The first order approximation for the dynamics of gait motion over the range of walking speeds gives us a fairly accurate set of invariant features, corresponding to the second harmonic normalized amplitude and phase terms (b2, ψ2) within figure 3.33. The simi- larity between the remaining harmonic coefficients is less accurate, since they are more dynamically related to the mode of motion and range of customary walking speeds. For this reason, it is better to use only the second harmonic terms within the feature vector of static gait parameters.
If we model the normalized limb angle functions θe(t) of both upper and lower leg seg- ments, then we have one parameter for the limb length ratio d2 and two parameters (b2, ψ2) for each modified Fourier series limb function. Overall, we have five distinct static parameters of gait motion that we can use for biometric identification. In addi- tion, if we model the hip joint displacement motionx(t) and y(t) with modified Fourier functions then the number of static parameters of gait motion can be increased to nine measurements. Although theoretically sound, further research needs to be done in order to validate the usefulness of these features to discriminate between subjects over the
customary range of different gait motions.