1. 0
the non-orthogonality rotation matrix expressed as | |
$^ 1l,
because for small angles, sin 8 = 8 and cos $ = 1, However,
this is not the Case for feature boundary matching. The
non-orthogonality may be large, for two reasons:
1l. As features grow, they may do so along non-orthogonal
axes.
2. As features rotate, the low resolution of the camera may
introduce several degrees of pseudo non-orthogonality.
Therefore, a rigorous solution Was utilized. Linearized
equations take the form :
AV + BA = £f
where A and B are the matrices of the partial derivatives
with respect to measurements and parameters, 4 is the vector
of the transformation parameters, and f is the discrepancy
vector.
In addition, a set of constraint equations was added to the
System. With the Proper flag set, the Solution could be
reduced to a five parameter (§ = 0), a four parameter (8 =
0, 5, = Sy), or a three parameter transformation (§ = 0, s,
= 8, = 1). The full System of equations is then:
AV + BA = f
CA = g
where C is the matrix of the partial derivatives with
respect to the constraint equations,
The boundaries were matched by considering row and column
pixel values as X and y point coordinates in the
transformation. The Software made an initial estimate of a
common point defining a local coordinate system origin in
both features to be matched, so that the boundary pixels in
each feature would have a common ordering. The boundary
fitting was iterated in order to improve the estimate of the
common point.
4 TESTS WITH A SINGLE ROTATED FILM
4.1 Data Collection
In order to test the boundary detection and feature
transformation algorithms, a single cranial X-ray was
selected and imaged four times. It was rotated
approximately 10, 15, and 25 degrees between successive
images, and also translated from 20 to 40 mm in X and Y. No
attempt was made to measure exactly the amount of rotation
Or translation. The series of images was designed to
simulate Sequential images of a growing person, except that
the features did not change in shape, only in location and
orientation. Also, the 'rotation angles were larger than
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