Methods:

In the appearance-based matching proposed by Cootes, a model of the grey-level variations is combined with an Active Shape model. For the former, PCA is used to reduce the dimensionality of the grey-level data and generate a linear grey variation model [2]:

$${\bf g = \bar g + P_gB_g}$$ (1)

where ${\bf\bar g}$ is the mean normalised grey-level vector, ${\bf P_g}$ is a set of orthogonal modes of variation and ${\bf B_g}$ is a set of grey-level parameters. In place of the 2D ASM we propose to use a 3D Warp Model, generated by statistical analysis of a large number of example deformation fields. To simplify computations, the 3D deformation vector fields are decomposed into volumes of orthogonal deformation components $x,y,z$. With PCA the linear warp variation model is expressed as:

$${\bf x = \bar x + P_xB_x;\;y = \bar y + P_yB_y;\;z = \bar z + P_zB_z}$$ (2)

Using the same notation as [2], this linear model allows any new warp instance ${\bf w(x,y,z)}$ to be approximated by ${\bf\bar w}$, the mean warp, ${\bf P_w}$, the set of orthogonal modes of warp variations, and ${\bf B_w}$, the set of warp parameters. The space of all possible elements expressed by eq.  is called the Allowable Warp Domain. Since there may be correlations between the grey-level and warp variations, grey-level and warp parameters are concatenated as follows

$${\bf B = [W_g'B_g'\; B_x'\; B_y'\; B_z']}$$ (3)

where ${\bf W_g}$ is a diagonal matrix of weights accounting for differences in dimensions between grey-level (intensity) and warp variations (distances). PCA of eq.  yields a super-set of parameters describing the complete appearance model

$${\bf B = QC}$$ (4)

where ${\bf Q}$ are appearance eigenvectors and ${\bf C}$ is a vector of parameters controlling both the warp and the grey-levels of the model. The core of the segmentation method consists then in matching a new grey-level image to one synthesized by the model using the appearance parameters. The iterative method described in [2] is used. After convergence, the solution explicitly contains warp variation parameters, which can be expressed back into $x,y,z$ components of the warp field and concatenated into ANIMAL vector format. Segmentation of the VOI is then possible using any structure model defined on the ANIMAL reference volume. It is achieved by applying the inverse of the deformation field to structures defined in the standard volume and then mapping those onto the subject.