ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

column vectors of P. Denoting the vanishing points in X , Y and Z directions as u, v and
w, the first three columns of P are u = p1 , v = p2 and αw = p3. The final column of P is
the projection of the world coordinate system in the image plane, i.e o = p4. The
vanishing lines between u, v is l with the constraint l• p1= l• u = 0 and l• p2= l• v = 0 .
The final column (the origin of the coordinate system) must not lie on the vanishing
line, since if it does, then all three columns are points on the vanishing line and thus are
not linearly independent. Hence o= p4= l/ l =l, resulting the parameterized projection
T
matrix P = [ u v αw
l] where α is the affine scale factor [1].
To measure distance between the scene planes with the points X=(X,Y,0,1) T and X ' =
(X,Y,Z,1) T in the scene planes, along with the images x and x ' in the image planes, π
T T
and π ' , we can write x = P(X Y 0 1) and x' = P (X Y Z 1) . Substituting P in the
expressions we get x = ρ (Xp1+ Y p2 + p 4)
and x' = ρ '(Xp1+ Yp2 + Z p3 + p 4)
where ρ
and ρ ' are unknown scale factors. Taking scalar product with l for x , we get
l•x = ρ(Xl•
p1+ Y l• p2+ l•l ) which reduces to ρ = l•x since l• p1= l• p 2 = 0 and
l•l=1. Now (Xp1+ Y p2 + p 4)
= x / ρ and putting this in the expression of x ' and
simplification and there after taking the cross product with x ' on both sides we obtain
x×x' α Z =−
( l•x) w×x' ……….(1)
The value of α in each reference direction can be determined form vanishing points u ,
v and w of an image along with the measured image point coordinates x and x ' and
known distance Z between corresponding space points X and X ' . Once it is found it can
be used for determining the distance Z for other points of interest, as its value will
remain constant for a particular camera setup.
5. EXPERIMENTAL FRAMEWORK
The experimental setup comprised of grabbing the images of palmer grasp sequence by
two cameras, one for palmer view ( Fig. 3a) and the other for lateral view( Fig. 3b) and
the radial distortion removed by method [6]. The reference plane in the experiment is π
and measurement plane is π ' . Each square grid in π is 176x176mm. A vanishing line
u is obtain by fitting a line between two user picked homogeneous image points
1
(a)
Fig. 3 (a) Image of palmer view for abduction-adduction (b) Image of lateral view for flexion-extension of
the same hand configuration.
(b)