Reconstruction error analysis

Each of the four test subjects are recorded walking at different speeds on the treadmill (Tunturi - J6). The set of captured sequences are then manually marked and the recon- struction process performed for values of Fourier harmonicsn= 1_{· · ·}10. The root mean squared reprojection error computed for each reconstruction gives us an indication of the goodness of fitting between the motion model and the imaged subject motion.

Figure 3.20:_{Root mean squared reprojection error generated from the gait recon-}

Figure 3.20 shows the means of these reprojection errors for the four test subjects, over the entire range of walking speed motion sequences. Three of the reprojection error curves are quite similar. The reprojection error starts quite high _≃ 2.8 pixels, by modelling the motion as simple harmonic, i.e. n = 1. This quickly falls below the

±1 pixel measurement error deviation, then slowly trails off to a level of_≃0.7 pixels. The reprojection errors corresponding to subject 01 level out at a value above the _±1 pixel measurement error, suggesting that here the model inadequately represents the dynamics of gait motion. There are a number of reasons why this level of error should differ from the others.

Care has been taken to ensure that all manually marked joint positions lie as close to the true positions as possible, to within a _±1 pixel tolerance. An analysis of the rigid articulated leg segment lengths over the range of walking speeds shows that the variation in limb lengths for subject 01 is much higher than in the others, illustrated in figure 3.21. This suggests that the markers may have been improperly placed for subject 01.

Figure 3.21: _{Analysis of the rigid articulated leg segment lengths over the range of}

controlled walking speeds.

Figure 3.22 shows the set of subjects in similar double stance postures. In this pose, the joint marker positions on the reference limb should be almost aligned in a single straight line. Natural gait alignment appears slightly flexed in this pose. Figures 3.22(a), 3.22(c) and 3.22(d) correspond well with this alignment while figure 3.22(b), corresponding to the pose of subject 01, shows a high degree of misalignment.

Accurate placement of markers is often quite difficult, especially over clothing. Markers are usually attached to a subject while they adopt a quiet standing posture. Since subjects remain still during the attachment procedure, markers taped over clothing retain their position. Clothing can tend to slip around during periods of locomotion in order to better fit with the body’s shape and motion, thus changing the positions of markers that were originally well placed. Experimenters often have difficulty attaching markers to certain positions of the body. Large markers within the hip area may be knocked by the swinging action of the arms. Baggy clothing can also allow the markers

(a) Subject 00 (b) Subject 01

(c) Subject 02 (d) Subject 03

Figure 3.22: _{Alignment of markers in similar double stance postures. Natural gait}

joint alignment should appear slightly flexed from the line of fit. This fitting line is computed via orthogonal regression through the set of joint markers. The alignment of subject 01 is significantly different from the norm, indicating that marker placement is poor.

to drift during motion. Experimenters must ensure that markers are taped firmly to each of the limb segments. On the other hand, skin and internal muscle structures need to be able to move in order to facilitate locomotion. Taping markers too firmly to joints causes stiffness within the limbs, and consequently subjects complain of unnatural walking motions.

Improper placement of markers also affects reconstruction. Placing markers over actual joint regions of the body is difficult since these areas undergo the largest changes in deformation. Figure 3.23 shows the geometric effect of placing a marker, in error, a small distance from the true joint position.

Figure 3.23:_{Effect of placing a marker a small distance away from the true joint}

position on the locus of motion.

The locus of motion of the marked knee pointX1 contains components of motion from both the upper and lower leg segments. Consequently, the lower limb lengthD2remains fixed while the upper limb lengthD1 varies with the phase of gait.

Table 3.3 shows the resulting root mean squared reprojection errors corresponding to the number of Fourier harmonics n used to model the limb motion. The total summation cost Pr, of the r.m.s. reprojection errors over all subjects and walking speeds, gives an indication of the ability to represent the dynamics of motion by using the required number of Fourier harmonics. The corresponding mean experimental r.m.s. pixel re- projection errorǫ=Pr/N, whereN is the total number of experiments, then gives an estimate of the level of pixel fitting error within any experiment.

n Pr ǫ ∆r ∆ǫ %error 1 33.4084 2.784 - - - 2 12.1011 1.0084 21.3073 1.778 63.7783 3 8.9985 0.7499 3.1026 0.2586 9.2868 4 8.6889 0.7241 0.3096 0.0258 0.9267 5 8.4443 0.7037 0.2446 0.020383 0.7321 6 8.362 0.6968 0.0823 6.8583_×10−₃ 0.2465 7 8.334 0.6945 0.028 2.3333_×10−₃ 0.0838 8 8.2992 0.6916 0.0348 2.9_×10−₃ 0.1042 9 8.2861 0.6905 0.0131 1.0917_×10−₃ 0.0391 10 8.2732 0.6894 0.0129 1.075_×10−₃ 0.0386

Table 3.3:_{Resulting root mean squared reprojection errors corresponding to the num-}

ber of Fourier harmonicsnused to model the limb motion. Reprojection errors are com- puted over all walking speeds and from all valid subjects in the experiments (Subjects 00, 02 and 03).

The reduction in r.m.s. fitting errors ∆r=Pr(n)₋Pr(n₋1), caused by increasing the number of harmonics used to represent the dynamics of gait, is also shown within table 3.3. The mean experimental error reduction ∆ǫ= ∆r/N then gives an indication of the reduction in pixel error within an experiment, caused by increasing the number

of Fourier harmonics. The percentage error reduction estimates 100_×∆r(n)/Pr(1) then correspond to the level of error reduction as a percentage of the simple harmonic experimental reconstruction errorPr(1).

We can clearly see that the quoted values of n = 5 represent a suitable choice for the number of Fourier harmonics required to model the limb motion. Here, the percentage error reduction estimate is less than 1% and any further increase in the number of harmonics n _≥ 6 used to represent the dynamics of motion have root mean squared pixel error reduction levels ∆ǫof the order 1_×10−₃

.