Model Space

Model space can be defined as the digital representation of the two physical volumes that can be processed by a computer. The links between physical and model spaces are provided by some measurement processes. Three factors limit the extent to which the physical volumes are accurately represented in model space.

• Quantization error: Only a finite number of discrete points of the volumes can be measured. Making the computer model continuous implies interpolation between the discrete data points. If the function describing a volume in physical space is not band-limited (which is certainly the case), the interpolation cannot be exact and will introduce errors in the representation of the volume.
• Geometrical distortion error (e.g., field inhomogeneity and flow for MR imaging, slice thickness for CT, pin-cushion for DSA).
• Measurement errors (noise) caused by imperfection of the measuring devices or by the incapacity of the user to precisely localize points in model space. This type of error is expected to be random.

The first goal of any Image-Guided neurosurgical procedure is to estimate the transformation between the volumes in physical space by using the imperfect information available in the model space. The estimated transformation will in general be different from because of the errors (mentioned above) introduced in the transformation from physical space to model space. Another error between and is introduced by the fact that the non-linear part of is not usually taken into consideration in the search for . In other words, is chosen to be the rigid body transformation that best matches . The error caused by this effect increases as the non-linear part of (i.e., the deformation between the two volumes) becomes more important. To take the non-linear part of the transformation into account would require the use of non-linear registration techniques [4,66,24,25]. We note here that the main factor contributing to the difference in shape of the patient's head between the scanner space the surgical space are swelling, displacement of tissues due to gravity (since the position of the patient is not necessarily the same in scanner and OR) and expansion or contraction of the brain after craniotomy due to pressure and drug-related effects.

It may not seem obvious why one should be concerned about the deformation of the skin since the region of interest is within the brain. However, registration must be performed before craniotomy and the points to determine the transformation matrix are generally situated on the skin.