In intramedullary nail (IMN) surgical operations, one of the main efforts for surgeons is to find the axes of two distal holes. Two distal holes on an intramedullary nail, which are inside the intramedullary canal of a patient's femur, can only be seen in a lateral X-ray view. For the standard surgical procedure, the localisation of the distal hole axes is a trial-and-error process which results in a long surgical time and large dose of X-ray exposure. In this paper, an algorithm to derive the 3D position and orientation of the distal hole axis was developed. The algorithm first derives the nail axis through two X-ray images. Then the distal hole axis is calculated through projecting back the hole boundary on the X-ray image from a lateral view to the 3D space. A least squares method is used to determine the centres of the front hole and the back hole through iteration. The algorithm has been tested with real data and it was accurate and robust.

This paper describes the mathematics of acquiring a cone axis through the projection of its conical boundary from X-ray images. A thorough analysis and rigorous proofs are presented for a better understanding of this basic problem. The results can be directly used for the trajectory planning of sliding hip screw and IntraMetullary nail operations by the computer assisted orthopaedics surgical systems.

It is common to use an affine transformation to approximate in dealing with the matching of plane curves under a projective transformation. The plane curve itself can be used as an identity to solve the parameters of an affine transformation. The objective of this paper is to obtain a closed form solution of the parameters using low order derivatives of the plane curve. A unique solution to the parameters of an affine transformation with up to second order derivatives is presented using differential invariants as well as the available global information. The computational time on verification has been significantly reduced. In computer vision, derivatives are obtained by numerical means. Achieving accurate numerical derivatives is an important application issue. Smoothing with a Gaussian filter modified by a linear combination of Hermite polynomials, can preserve the accuracy of continuous polynomials with powers up to the same order as the Hermite polynomials. In the discrete space however, the introduction of Hermite polynomials leads to a choice of a large smoothing scale /spl sigma/ in order to reduce computational errors at the expense of a reduction of local controllability and over-smoothing of the curve. It is shown that using a /spl sigma/ proportional to the order of the derivatives is more reliable in applications.

The invariant used as an index has shown many advantages over the pose dependent methods in model-based object recognition. Although perspective and even weak perspective invariants do not exist for general three dimensional point sets from a single view, invariants do exist for structured three dimensional point sets. However, such invariants are not easy to derive. A new special structure for calculating invariants of three dimensional objects is presented. The 3D invariant structure proposed by Rothwell (1993) requires seven points that lie on the vertices of a six-sided polyhedron and is applicable to position free objects. In comparison, the proposed algorithm requires only six points on adjacent (virtual) planes that provides two sets of four coplanar points and does not require the position free condition. Hence it is applicable to a wider class of objects. The algorithm is demonstrated on images of real scenes.

The invariant used as an index has shown many advantages over the pose dependent methods in model-based object recognition. Although perspective and even weak perspective invariants do not exist for general three dimensional point sets from a single view invariants do exist for structured three dimensional point sets. However, such invariants are not easy to derive. The 3D invariant structure proposed by Rothwell (1993) requires seven points that lie on the vertices of a six-sided polyhedron and is applicable to position free objects. A new special structure for calculating invariants of three dimensional objects is developed by the authors (1995). In comparison, the proposed algorithm requires only six points on adjacent (virtual) planes that provides two sets of four coplanar points and does not require the position free condition. Hence it is applicable to a wider class of objects This paper is the extension of previous work to discuss how to use the projection to the base plane to obtain invariant conditions for the more general situation. The algorithm is demonstrated on images of real scenes.