3.6 Treadmill experiments
3.6.7 Intra and inter-class variation
3.6.7.1 Discrimination between individual parameter features
We first investigate the variation within individual parameters in the proposed biometric feature vector P. We therefore assume that the feature vector only contains a single parameter, thus the distance metric ρ=kPb −Pk describes the intra and inter-class variances,σ2
v andσc2, for that parameter. Furthermore, we can quantitatively assess the
ability of this parameter to discriminate between people, by evaluating two properties that are dependent on these class variances.
The class distinction quantity γ describes the percentage of the intra-class variance compared with that of the inter-class variance, i.e. γ = 100×σ2v/σc2. Low percentages indicate good discrimination between different subjects, while high percentages highlight the inability of the biometric parameter to distinguish between people at all.
The class distinction says nothing about the magnitude of the parameter variation, only the ratio of variation between classes. The parameter distinction quantity β describes the intra-class parameter deviation as a percentage of the mean subject parameter es- timate µp, i.e. β = 100×σv/µp. This percentage gives us an indication of the relative
magnitude of the intra-class parameter deviation.
Normalized limb segment lengths. The variance of the normalized limb segment lengthsdj is examined over the set of biometric parameter vectorsPi. The gait distance
metric ρ=kPb−Pk, corresponding to the normalized limb segment lengths dj, can be
written. ρ2 = m X j=2 ³ b dj−dj ´2 (3.30)
Figure 3.40 shows the difference matrix D corresponding to the normalized limb seg- ment length parameterd2. The normalized limb lengths can be described as the static geometric parameters of gait motion, which remain invariant to any changes in walking motion. Consequently, we expect the variance within these parameters to be quite low.
Figure 3.40: Difference matrixDcorresponding to the normalized limb segment length
parameter d2. Intra and inter-class variances: max = 0.224398, intra = 6.3693e-05,
inter = 0.003951,γ= 1.61%,β= 0.91%.
The figure confirms this prediction, and shows good distinction between different sub- jects. The intra-class deviation of parameter d2 is estimated at 0.91% of the mean subject limb segment length, and the magnitude of the intra-class variance at 1.61% of the inter-class variance level. This demonstrates that d2 is a well defined biometric parameter that has good discrimination between subjects.
Normalized amplitude components. The variance within each of the normalized amplitude components bk is examined over the set of biometric parameter vectors Pi.
The corresponding gait distance metricρ=kPb−Pk has the form:
ρ2= n X k=2 ³ bbk−bk ´2 (3.31)
Figure 3.41 shows the difference matricesD corresponding to the individual normalized amplitude componentsbk of the leg angle function (components of both upper and lower
limb segments). We have previously shown within section 3.6.4, that the intra-class variance of the second order amplitudesb2 is reasonably static.
The intra-class deviation of parameter b2 is estimated at 3.78% of the mean normalized amplitude, and demonstrates that the coefficientb2 remains relatively constant over the range of customary walking speeds.
The magnitude of the intra-class variance is estimated at 57.64% of the inter-class vari- ance level. The intra-class variance then has almost twice the level of discrimination
(a)b2: max = 0.082147, intra = 0.000211,
inter = 0.000365,γ= 57.64%,β= 3.78%
(b)b3: max = 0.099194, intra = 0.000373,
inter = 0.000534,γ= 69.7%,β= 20.96%
(c)b4: max = 0.034871, intra = 6.03734e-05,
inter = 8.96055e-05,γ= 67.38%,β= 29.55%
(d)b5: max = 0.024746, intra = 2.01419e-05,
inter = 3.36887e-05,γ= 59.79%,β= 19.95%
Figure 3.41: Difference matricesD corresponding to the individual amplitude com-
ponentsbk of the normalized leg angle function.
over the inter-class matches. As a biometric feature, the normalized amplitude com- ponent b2 appears relatively weak in comparison to the geometric static parameter of gait d2, which has over sixty times the level of discrimination.
The higher order amplitudes are less reliable and show significant levels of parameter deviation β > 20% from their mean estimates. Discrimination between the intr and inter-class variances is also poor, withγ >60%. Table 3.4 shows the result of combining ranges of normalized amplitudesbkwithin the distance metricρ. The distinction between
class variances appears to worsen as more components are combined. The table suggests that we are not able to distinguish between people at all if we include these higher order components.
P
k Intra-class variance Inter-class variance γ
k= 2 0.000210425 0.000365082 57.64%
k= 2· · ·3 0.000500881 0.000503108 99.56%
k= 2· · ·4 0.00052302 0.000467122 111.97%
k= 2· · ·5 0.000518958 0.000444299 116.8%
Table 3.4: Discrimination over all normalized amplitude componentsbk. Combining
the higher order amplitude components results in equal intra and inter-class variances.
Normalized phase components. The variance within each of the aligned phase components ψk is examined over the set of biometric parameter vectors Pi. We first
study the variance of the dot product between unit vector representations of these phase angles. The gait distance metric ρ=kPb −Pk can then be written in terms of the corresponding phase angle unit vectors bvk andvk.
ρ2= n X k=2 µ 1 2(1−vb ⊤ kvk) ¶2 (3.32)
Figure 3.42 shows the difference matricesDcorresponding to the individual phase com- ponents ψk of the normalized leg angle function (components of both upper and lower
limb segments). Figures 3.29 to 3.32 on page 97 showed that the lower order intra- subject phase angles remained fairly consistent over the range of walking speeds. This is also reflected by the quantitative assessment of the intra-class variance shown within figure 3.42. The dot product measure between phase vectors lies within the range (0 : 1), therefore we express the parameter distinction quantity β as the percentage of intra-class deviation over this unit range, i.e. β = 100×σv. This percentage devia-
tion remains quite low≃ 2% for the first two phase components then quickly becomes unstable. This is also reflected by the corresponding poor class distinction percentages γ between the intra and inter-class variances. The higher order components have equal intra and inter-class variances, thus are unsuitable as potential biometric features. As biometric parameters, the phase components are similar to the normalized ampli- tudes, in that they appear relatively weak in comparison to the geometric static param- eter of gait d2. This is not unexpected, since leg motion is highly dynamic and we have only approximated a set of consistent gait features over the range of walking speeds. The level of variance within each phase component increases with higher order. We can then choose to weight the contribution of each phase component with the corre- sponding normalized amplitude, in order to increase the significance of the lower order phases. Subsequently, the magnitude weighted phase version of the gait distance metric ρ=kPb −Pk can be defined as:
ρ2 = n X k=2 µ bk 2(1−vb ⊤ kvk) ¶2 (3.33)
(a)ψ2: max = 0.130181, intra = 0.000412, inter = 0.000901,γ= 45.69%,β= 2.03% (b)ψ3: max = 0.156812, intra = 0.000572, inter = 0.001564,γ= 36.58%,β= 2.39% (c)ψ4: max = 0.883111, intra = 0.093758, inter = 0.069765,γ= 134.39%,β= 30.62% (d)ψ5: max = 0.32412, intra = 0.009619, inter = 0.008877,γ= 108.37%,β= 9.81%
Figure 3.42: Difference matricesDcorresponding to the individual phase components
ψk of the normalized leg angle function. No magnitude weighting is performed within the distance metricρ.
Table 3.5 compares the class distinction percentages γ between magnitude weighted phase and normal phase variances. The class distinction between higher order phase components is improved by the magnitude weighting, while the lower order components remain similar.
k Intra-class variance Inter-class variance γ (MWP) β γ 2 2.50272e-05 6.23271e-05 40.15% 2.03% 45.69% 3 1.09812e-05 2.48628e-05 44.17% 2.39% 36.58% 4 2.44339e-05 4.06368e-05 60.13% 30.62% 134.39% 5 2.02181e-06 6.00553e-06 33.67% 9.81% 108.37%
Table 3.5:The magnitude weighted phase (MWP) components ψk of the normalized
limb angle function. The class distinction between higher order components is improved by the magnitude weighting.
Table 3.6 shows the result of combining ranges of magnitude weighted phase components within the distance metric ρ. The first two phase components ψ2 and ψ3 individually remain relatively constant over the range of walking speeds. The overall distinction between intra and inter-class variance is then improved by combining both of these phase components, for both non-weighted and magnitude weighted distance measures.
P
k Intra-class variance Inter-class variance γ (MWP) γ
k= 2 2.50272e-05 6.23271e-05 40.15% 45.69%
k= 2· · ·3 3.17207e-05 8.04965e-05 39.41% 39.82%
k= 2· · ·4 4.19072e-05 8.8079e-05 47.58% 139.57%
k= 2· · ·5 4.18024e-05 8.7462e-05 47.79% 172.11%
Table 3.6:Discrimination over all magnitude weighted phase (MWP) components.
Individually, the higher order phase components ψ4 and ψ5 vary significantly within their unit range, and have similar intra and inter-class variances. Consequently, the level of class distinction is worsened by combining these higher order phase components within the distance metric. Table 3.6 shows that the use of magnitude weighted phase clearly alleviates the impact of including these noisy higher order phase terms.