Anatomical Information, Intensities and Bias Field

For a model-based segmentation method, the anatomical information is encoded in a model of the anatomical shapes. This is represented by $S$ in this formulation and in practice would take the form of something like a mesh model of the shapes of interest (e.g. ventricles and deep brain structures for our neuroimaging application).

The measured image, $Y$, is related to the shape model in two ways: by a spatial transformation, $T$; and by an intensity generation model, $G$. By adding prior information to what spatial transformations are more likely, the typical variation in the shapes across a population can be encoded. That is, $p(T)$, encodes the information about modes of shape variation (typically by using a multi-variate Gaussian).

The intensity generation model relates the (noise-free) intensity values to the spatially transformed shapes. This generation model includes information about how partial volume is generated during imaging.

For MRI, the actual intensity values are related in a complicated way to the tissue parameters, which is controlled by the pulse sequence, allowing many different image `contrasts' to be formed. Consequently, without knowledge of the pulse sequence details, the intensities within each shape can take on a wide range of possible values, and so are modelled by independent parameters, $\alpha_{m,k}$.

In addition, MR images often contain bias field. This is an effect of inhomogeneities in the RF field, and usually effect both transmit and receive signals. The effect is to have a slowly varying intensity change across the image.²Often this is modelled as a slowly varying multiplicative change, but because the bias field also effects the trasmitted RF, there can be sharp changes between its effect at tissue boundaries, due to differing flip angles causing changes in the steady-state magnetisation. Therefore we choose to model this effect by including a linear spatially-varying intensity for each shape, which is an adequate approximation for shapes of small spatial extent relative to the effective wavelength of the bias field, and allows both effects of the bias field to be incorporated. Furthermore, such linear terms also model underlying spatial variations in the tissue parameters, which can exist (e.g. changing density within the thalamus) and give rise to spatially intensity changes in the image which are indistinguishable from bias field effects.