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Statistical Models of Appearance

Statistical models of shape and appearance (combined appearance models) were introduced by Cootes, Edwards, Lanitis and Taylor [1,2,7], and have since been applied extensively in medical image analysis [9,13,17]. The construction of an appearance model depends on establishing a dense correspondence across a training set of images using a set of landmark points marked consistently on each training image.

Using the notation of Cootes [2], the shape (configuration of landmark points) can be represented as a vector $\mathbf{x}$ and the texture (intensity values) represented as a vector $\mathbf{g}$.

The shape and texture are controlled by statistical models of the form


\begin{displaymath}
\begin{array}{cc}
\mathbf{x}=\mathbf{\overline{x}}+\mathbf{P...
...}=\overline{\mathbf{g}}+\mathbf{P}_{g}\mathbf{b}_{g}\end{array}\end{displaymath} (2)

where $\mathbf{b}_{s}$ are shape parameters, $\mathbf{b}_{g}$ are texture parameters, $\mathbf{\overline{x}}$ and $\overline{\mathbf{g}}$ are the mean shape and texture, and $\mathbf{P}_{s}$ and $\mathbf{P}_{g}$ are the principal modes of shape and texture variation respectively.

Since shape and texture are often correlated, we can take this into account in a combined statistical model of the form


\begin{displaymath}
\begin{array}{cc}
\mathbf{x}=\bar{\mathbf{x}}+\mathbf{Q}_{s}...
...\mathbf{g}=\mathbf{\bar{g}}+\mathbf{Q}_{g}\mathbf{c}\end{array}\end{displaymath} (3)

where the model parameters $\mathbf{c}$ control the shape and texture simultaneously and $\mathbf{Q}_{s}$, $\mathbf{Q}_{g}$ are matrices describing the modes of variation derived from the training set. The effect of varying one element of $\mathbf{c}$ for a model built from a set of 2D MR brain image is shown in Fig. 1.

\includegraphics[%%
scale=0.4]{../Graphics/EPS/brain_0_cps.eps}

Fig. 1. The effect of varying the first model parameter of a brain appearance model by $\pm2.5$ standard deviations.


next up previous
Next: Model-Based Evaluation Up: Background Previous: Assessing Non-Rigid Registration
Roy Schestowitz 2007-03-11