In this section a new method of computing the projection matrices of an image sequence will be presented.
A video camera can be used to obtain a long sequence of images. The motion between successive frames will be small but the overall motion of the viewer will be large. The trifocal tensor of the three most extreme views will be much more accurate than the tensor of three successive views. Once this outer tensor is known, the intermediate projection matrices can be computed using the linear algorithm which will now be presented.
The projection matrices of two views can be obtained from the initial tensor. These two matrices are represented by a normalised camera matrix P=[I | 0] and a 3 x 4 projection matrix P', then the object is to compute for each view in turn along the sequence the 3 x 4 camera matrix P''.
The equation for obtaining the trifocal tensor from three camera matrices where the first camera is normalised Hartley95 is:
Since the entries of P'rc are known, the vector t , containing the entries of Tijk written as a 27 x 1 column vector, can be written as a linear function of b , which is a 12 x 1 column vector containing the entries of P'' .
In this equation H is a sparse 27 x 12 matrix containing the entries of P' .
The linear solution to the trifocal tensor can be written as a measurement matrix M multiplied by the tensor vector t , and by combining equations 15 and 16, a linear solution to the third camera matrix is obtained.
The linear solution to this equation gives the entries of the new camera matrix b .