3.6 Treadmill experiments
3.6.1 Limb angle function
We have previously shown that the motion of a single articulated limb is approximately planar. A good initial estimate for this limb swing plane can be computed by aligning a calibration grid pattern with the subject’s leg on the treadmill, while in the quiet standing posture. Figure 3.18 shows the configuration of a calibration grid aligned with a subject’s leg plane.
Figure 3.18: Vertical reference plane of the subject’s leg.
The projection of metric points u from the canonical reference plane to the imaged leg swing plane can be computed by the planar homography mapping as ˆx=Hu. The correspondence between known grid pointsu on the metric plane and the set of imaged points x allows us to compute the elements of the homography matrix H by solution of the Direct Linear Transformation [x]×Hu=0. These equations can be rearranged
into the formA·h=0, whereh is the flattened set of homography matrix coefficients. Further details of the Direct Linear Transformation can be found in section 2.4.6. The putative set of limb joint positions w on the metric plane are first recovered by applying the inverse mapping w=H−1
x to the set of imaged marker points x. The set of articulated leg segments are rigid and have fixed lengths over all frames in the sequence. We can accumulate the vector d, of mean upper and lower leg segment lengths over the image sequence, by computing the distances between recovered joint marker endpoints. Since we are only interested in limb length ratios, we compute the normalization transformation Kn, that maps the centroid of all joint marker positions wover the sequence to the origin, with isotropic scaling such that the first limb segment has unit length. The set of metric plane points are then transformed as w′
=Knw. Consequently, the updated homography that maps these normalized points into the image is given by H′
=H·Kn−1. Angles between joint marker endpoints are invariant to the similarity transformation Kn, hence the transformation serves as a convenient way to normalize the data into a consistent format over the set of test subjects.
We first need to find an initial estimate of the fundamental frequency of gait f0. A cost function, that is dependent on the self-similarity of limb direction vectors between putative values of gait period, is then a good indicator of the subject’s periodicity. The dot productCbetween limb angle unit vectors is maximal (unity) when the two vectors are similar. The cost 1− C is then minimal between similar positions of pose. A vector of root mean squared self-similarity costs is then accumulated over a suitable range of putative periodicities. We know the camera frame rate accurately and can use the 1 Hz estimate of natural gait to determine this sensible range of putative periodicities. To eliminate any false local minima caused by measurement noise within the cost vector, we first apply a (1,4,6,4,1) smoothing filter to the vector of self-similarity cost errors. We then assume that there is a single minima within the chosen range of putative peri- odicities that represents the true period of subject gait. The true (discrete) periodicity estimate is found by performing gradient descent on the vector of self-similarity cost errors from a 1 Hz initial periodicity estimate. A sub-temporal estimate of the period T is then found by fitting a quadratic curve to the data about this computed discrete periodicity estimate. A more detailed discussion on periodicity detection is given later within chapter 4, as part of the gait reconstruction algorithm.
The set of valid limb angles and their corresponding time sample vectors are computed for each of the normalized leg segments. The Fourier series representation of each limb angle function is then determined, with fixed fundamental frequency f0= 1/T. The set of minimized coefficients for each limb segment are then stored in a biometric recon- struction vectorVi, where the coefficients of V have the form:
V= (a0, a1, φ1,· · ·, an, φn)
⊤
(3.9)
With the knowledge of the normalized leg lengthsD, we can find by back substitution the best set of hip points X0 consistent with the limb angle functions.
Figure 3.19:Articulated limb segment model. The hip pointX0is defined by a set of
Cartesian (x, y)⊤
coordinates. The remaining articulated limb endpoints are defined by a connected set of polar coordinates (d, θ)⊤
. The first limb segment length is canonically normalized to unit length.
Figure 3.19 shows the model of articulated limb connections. The hip pointX0 is given by the Cartesian coordinates (x, y)⊤
while the remaining limb endpoints are defined by the connected set of polar coordinates (d, θ)⊤
, where the first limb segment has been normalized to unit length. The Cartesian coordinates (xi, yi)⊤ of any limb point with
indexiin the model is then given by the equation:
(xi, yi) ⊤ = (u, v)⊤ i= 0 (u, v)⊤ +Pij=1Dj ·(sinθj,cosθj)⊤ i≥1 (3.10)
where the pose angles θj are given by evaluating the Fourier series function θ(t) at
the current pose frame with the biometric coefficients Vj and fundamental frequency
f0 = 1/T.
Given any endpoint in the articulated limb set, we can compute the putative position of the hip point (u, v)⊤
by back substitution. Since a limb segment endpoint is com- puted relative to its predecessor, measurement fitting errors will be compounded within the back substitution process. A weighted putative hip point (u′
i, v ′ i, w ′ i) ⊤ , where w′ i is
the associated weighting factor, is computed from each of the valid metric plane data points (˜xi,y˜i)⊤ of the articulated leg pose. The putative hip point, from any indexed
metric plane data point i= 0· · ·m within the current leg pose, is given by the set of equations. (u′i, v′i, w′i)⊤= (m+ 1)·(˜xi,y˜i,1)⊤ i= 0 (m+ 1−i)·h(˜xi,y˜i,1)⊤−Pij=1Dj·(sinθj,cosθj,0)⊤ i i≥1 (3.11) Where m is the total number of segments within the articulated leg model, i.e. two for a model of upper and lower legs. The fitted hip point (u, v)⊤
is then given by the summation of all valid weighted points.
(u, v)⊤= Ã Pm i=0u ′ i Pm i=0w ′ i , Pm i=0v ′ i Pm i=0w ′ i !⊤ (3.12)
We only require a minimum of one metric plane data point within a leg pose to compute the associated hip point. This resolves the problem that arises when the swinging arm occludes the hip point marker. The set of computed hip points Xi on the metric plane
3.6.2 Maximum likelihood estimation
We have computed an initial estimate of the set of parameters that model the articulated leg motion over the image sequence. As a final step, we optimize the motion parameters Pin order that we minimize image reprojection error. The parameter vector P can be partitioned into two sections. The first contains the set of coefficients common over the entire sequence (h,D, f0,V). The second contains the subsidiary set of instantaneous hip positionsXi = (ui, vi)⊤ corresponding to each of the individual frames.
P=³h⊤,D⊤, f0,V
⊤
|X⊤1,· · ·,X⊤N´
⊤
(3.13)
Where h is the vector of homography coefficients that map points on the metric plane into the image, D is the vector of normalized limb lengths, f0 is the fundamental fre- quency of gait and V contains the sets of articulated limb segment Fourier coefficients. The set of parameters P are optimized by performing the Levenberg-Marquardt mini- mization method. The form of the Jacobian is sparse and consequently the minimization procedure is similar to that described within appendix C.4.