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1995 
  
333 
Substituting Equation (7) into Equation (6), the following linear equation is obtained with the rigid motion parameters 
as unknowns 
(5x FF) R+E -T+F=0. (8) 
If the above equation is derived for n points on the rigid object (where n is not less than the number of unknowns), 
the motion parameters are obtained as a least square solution of the system of linear equations. The range flow 
can be calculated by substituting the obtained motion parameters into Equation (8). In order to guarantee a unique 
solution i.e., unique motion interpretation, it is necessary that the system of equations be linearly independent. 
3.3 Intensity Constraint of Motion 
In practice, there are several objects from which rigid motion parameters cannot be uniquely determined. Equation 
(6) is related to the surface structure of the object. The equation is strongly constrained by depth information. In 
return, in the case where the surface of the object is nearly planar, this constraint has a decreased sensitivity to the 
detection of motion. If one can utilize intensity information in addition to the surface structure, so that the ambiguity 
of motion interpretation is less sensitive to ill-posed conditions; then, the accuracy of motion parameters will improve. 
Let the intensity of the image at point P on the object at time £ be E(z, y,t). The range flow Ap of the point P is 
known to be constrained by 
E,-Ap+E =0 (9) 
where É, = (E,, Ey, 0)", (Ex, Ey) is a tangent gradient of the intensity image, and E; is the difference between 
the successive intensity images [7]. 
Substituting Equation (7) into Equation (9), one can obtain the following motion equation: 
(x É).E. E, T E-O (10) 
However, even if a lone constraint of the depth or the intensity cannot produce a unique motion interpretation, both 
constraints together may provide a unique motion interpretation. 
4. ALGORITHM IMPLEMENTATION 
In the previous paragraphs, a method to calculate rigid motion parameters from range and intensity image sequences 
has been described. Implementation of this method in a computational algorithm will now be discussed. 
4.1 Differentiation of Range and Intensity Images 
A tangent gradient of both range and intensity images is calculated using the Sobel edge operator. A temporal 
difference between images is obtained from a difference between average values in a 3 x 3 window of successive 
images. But since most images have a high level of noise compared to high resolution slower range sensors [15], 
one needs to filter them before differentiation occurs. In this implementation one uses the intrinsic filter described 
in [6] to reduce the effect of noise. This filter has the advantage of being invariant to viewpoint and is capable of 
preserving sharp variations. 
4.2 Solving the System of Constraint Equations 
The motion parameters can be obtained from solving the system of linear equations, which consists of two classes of 
constraints i.e., depth expressed by Equation (8) and intensity expressed by Equation (10). The relative importance of 
each constraint equation is determined by an appropriate constant weighting value w; so that the system of equations 
does not fall into an ill-posed condition. 
The solution of the system of equations is, in general, inaccurate since Equations (6) and (9) are only first order 
approximations. In order to obtain a more accurate solution, an iterative calculation is usually required, which is 
similar to the multidimensional Newton's method. 
A range flow of points (x,y,z) is assumed to be estimated as (Az, Ay, Az), which is given by substituting the 
obtained motion parameters into Equation (8). If the range flow is valid, it should be F(z,y,t) = F(z + Az, y+ 
Ay,t+ At) — Az and E(z,y,t) = E(z + Az, y+ Ay,t + At). Otherwise, one can define a new range and intensity 
image as F'(z,y,t + At) = F(z + Az, y+ Ay,t + At) — Az and E'(x,y,t + At) = E(z + Az, y+ Ay, t + At), 
respectively. Then, one can calculate the motion parameters using the new images until the residual square-error 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995