Carter, John 
  
Measures of angle behave well under perspective transformation, retaining many of their properties regardless of the 
orientation from which they are observed. In the following section we will outline the basis of a simple geometric 
correction and show its efficacy experimentally. Subsequent to that, we will show how simple notions are inadequate 
when applied to a real walker and will develop a more sophisticated model. In the final section of the paper this leads to 
the notion of a pose invariant gait signature. 
3  GAIT SIGNATURES AND THE TRAJECTORY ANGLE 
As outlined above, the fundamental basis of model based gait analysis is the measured angle, in particular the hip 
rotation angle. As in any computer vision application, angles are determined using the inverse arc tangent applied to 
measured horizontal and vertical components. Thus in a calibrated camera system, under an orthographic projection, an 
angle can be computed correctly from simple distances measured in pixels. This is should also be valid under the 
perspective projection, if the object under scrutiny is sufficiently far from the camera. These are the conditions that hold 
for most laboratory investigations of gait analysis, where the subjects parade at right angles to the camera. However, in 
the real world, subjects under investigation will typically be viewed from an oblique angle, i.e. the camera may be 
looking down onto the subjects and the subject may be walking diagonal to the camera view, Figure 1. Let us consider 
these two cases separately and independently. 
3.1 Elevation Angle 
  
Consider a typical surveillance system where the camera system 
T is mounted some height above the ground and is inclined 
To Camera H | towards the ground. The angle made by the optical axis of the 
camera and the ground plane is hereafter known as the elevation 
H z angle. Now consider a set of hip rotation measurements made as 
O 
  
  
  
  
  
9 T a subject walks across the field of interest and at right angles to 
the projection of the optical axis onto the ground plane. In an 
K A.V Measured ideal system (neglecting calibration, distortion and perspective) 
K or Gait Angle the single effect of the elevated position on the component 
Angle measurements of the rotation angle, will be a foreshortening of 
the vertical component by the cosine of the elevation angle. 
(a) Plan View (b) Camera View Interpreting these measurements, in terms of a human walker, 
suggest that in any realistic measurement system the vertical 
Figure 2 Looking down on the plan view (a) the component will be nearly constant and the angular information 
positions of the Hip, H, and the Knee, K, are will be carried almost entirely in the horizontal component. This 
seen relative to the origin O. The vector HK suggests that, within reason, the elevation angle will have no 
defines the walking direction, and thus the significant bearing upon the determination of the true hip 
trajectory angle 0. The view from the camera (b), — rotation angle. More specifically, if ¢ is the true hip rotation 
shows the gait angle ¢ in relation to the angle, y is the measured hip rotation angle and £ is the elevation 
coordinate svstem. angle, and if small angle approximations hold we can write 
tona Sut or v= sin(®) : a) 
cos(®)cos(e) cos(d) cos(e) 
The interpretation of this equation is simply that at most the elevation angle will contribute a constant scaling factor to 
the measured gait angle and as such we will ignore its effect hereafter. Of course, in the limit, as € tends to 90 degrees 
this will break down. 
3.2 Trajectory Angle 
Consider now the second case, where the camera is viewing the walking subject with the optical axis parallel to the 
ground plane. Furthermore, consider the effect of determining v if the subject is walking at an angle to the axis of the 
camera, we call this angle the trajectory angle, 0, see Figure 2, where H is the hip and K is the knee. In a similar manner 
to the elevation angle, the trajectory angle will manifest itself by foreshortening the horizontal component of the gait 
angle, suggesting that the equation 
iS sin($) cos(0) . tan(y) 
tan(y) cond) or tan(¢) = cos(0) X2) 
  
116 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 
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