The foundations of computer assisted surgery

Using navigation systems in general orthopaedic surgery and, in particular, knee replacement is becoming more and more accepted. This paper describes the basic technological concepts of modern computer assisted surgical systems. It explains the variation in currently available systems and outlines research activities that will potentially influence future products. In general, each navigation system is defined by three components: (1) the therapeutic object is the anatomical structure that is operated on using the navigation system, (2) the virtual object represents an image of the therapeutic object, with radiological images or computer generated models potentially being used, and (3) last but not least, the navigator acquires the spatial position and orientation of instruments and anatomy thus providing the necessary data to replay surgical action in real-time on the navigation system's screen.

Foundations of stochastic geometry and theory of random sets

The first section of this chapter starts with the Buffon problem, which is one of the oldest in stochastic geometry, and then continues with the definition of measures on the space of lines. The second section defines random closed sets and related measurability issues, explains how to characterize distributions of random closed sets by means of capacity functionals and introduces the concept of a selection. Based on this concept, the third section starts with the definition of the expectation and proves its convexifying effect that is related to the Lyapunov theorem for ranges of vector-valued measures. Finally, the strong law of large numbers for Minkowski sums of random sets is proved and the corresponding limit theorem is formulated. The chapter is concluded by a discussion of the union-scheme for random closed sets and a characterization of the corresponding stable laws.