Orthogonal and least-squares based coordinate transforms for optical alignment verification in radiosurgery

Radiosurgery is a non-invasive treatment technique applying focused radiation beams. It requires high geometric accuracy as misalignment can cause damage to the surrounding healthy tissues and loss of the therapeutic effect. One promising technique to insure submillimeter alignment accuracy of the radiation beam is to optically monitor the position of the beam axis relative to a frame firmly attached to the patient's skull using an optical alignment system. The optical alignment method requires three-dimensional coordinate transforms. This paper compares the standard least squares technique for transforming the coordinate system with an orthogonal transform technique based on the comparison of marker triangles. Orthogonal transforms have good numerical properties and preserve distance, which gives this technique advantages over non-orthogonal techniques. The mathematics behind each technique is covered and alignment results on real test equipment are used to illustrate the differences between the transforms.