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be calculated by conversion to homogeneous co-ordinates and multiplying by a standard y-axis rotation matrix giving 
position K' as 
cos(O) 0 sin(@) O||/sin($) x 
Qus 1 9 e| [oso |» 
K- —sin(6) O cos(0) 0 0 T z 
0 0 0 1 1 0 
-|cos(0)/sin($) -x 1cos(b)+y —sin(0)Isin(()x+z 1] 
, (4) 
H'=H 
Applying the perspective transformation, the positions of the hip and knee can be calculated by the screen co-ordinates 
of a simple pinhole camera that is D units away from the walker and has a focal length of P units. From the transformed 
co-ordinates above, the apparent angle to the vertical y can be calculated from 
  
where subscripts denote the x, y or z components of the vectors H' and K’, giving 
  
  
P P 
x — (cos(6)/ sin(d) + devis 
an) = | D- Ji Be sin(0) —z | (6) 
b tes +sin(0)/sin(d) x 
If the camera is assumed to be far away from the walker, and D is always greater than components of position, then x, y 
and z can be neglected and P, D and / cancel, so the equation for the measured angle then simplifies to 
any) OT, qf 
or more usefully 
| Sin(y) 
a cos(0)cos(w) - G) 
Equation 9 implies that once the trajectory angle is known, then the true hip angle can be calculated directly from the 
measured angle. Furthermore, in the limit of small angles, the correction is simply a linear scaling by cos (q). Thus if 
gait trajectories are normalised to correct for natural variations in amplitude i.e. walking speed, then this pose correction 
is unnecessary. 
The theory behind Equation (8) is easily extended to account for an inclined leg swinging plane, and can be 
reformulated as 
simu d lan. (9) 
cos(y) cos(0) cos(®) 
Here a is the leg angle described at the end of Section 3. above. Note that while the right hand side of Equation (9) is 
not independent of the measured angle, a unique solution for tan(£) does exist. Detailed calculations indicate that over 
the range of gait angles for ordinary walking, approximating cos(£) as unity is acceptable. This approximation has been 
used to generate the data plotted in Figure 9. The different gait curves overlap significantly and any residual differences 
are interpreted as experimental errors and limitations in the 4* order Fourier Series as a description of human gait. As 
such, a pose independent metric has been achieved by incorporating scene geometry. 
  
tan(@) = 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 119