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Appearance Model Construction

The parameters which statistically describe the shape (much as in active shape models) can be expressed as a vector $x$, where

$$x=x_{mean}+P_{s}b_{s}.$$ (1)

$x_{mean}$ (or $\overline{x}$) is the mean shape, as was calculated from the training data using, for example, Procrustes analysis³. $P$ represents the eigenvectors of the covariance matrix (set of orthogonal modes of variation) and the parameters $b_{s}$ control the variation of the shape. For $n$ modes of variation, $1<s<n$ holds.

Similarly, a vector $g$ is used to describe the intensity of given pixels as derived from axis-aligned input data that is stretched to encompass the whole shape and fit or overlap the original model dimensions. Usually warps are used to displace the control points until they match those of the mean shape and shape-normalised patches can be captured. Just as before, variation is subjective to

$$g=g_{mean}+P_{g}b_{g}.$$ (2)

For shape, training is affected especially by the choice of landmarks identified in the image, whereas to extract intensity values a different approach is in use. This approach relies on the fact that geodesic interpolation can be applied to compensate for the noncontinuous results of the triangulation algorithm used. The linear form of the model as expressed above (1)(2) is due to Principal Component Analysis (PCA) which reduces the length of the vectors describing shape and texture, namely $x$ and $g$ respectively.

It is now imperative that the two equations above are merged in some way to create a new model that captures both shape and intensity. To do so, $b_{s}$and $b_{g}$are aggregated so they can be expressed as one single column vector

$$\left\{ \begin{array}{cc} b_{s}\\ b_{g}\end{array}\right\} .$$ (3)

Applying further PCA, the following model is obtained:

$$\begin{array}{cc} x_{i}=\bar{x}+Q_{s}c_{i}\\ g_{i}=\bar{g}+Q_{g}c_{i}\end{array}.$$ (4)

It is purely controlled by $c_{1},c_{2},...,c_{n}$ where $n$ is intended to be smaller than the number of $b_{s}$ and $b_{g}$ combined. That is simply due to the dimensionality reduction of PCA. Usually an inclusion of some weighing W is included to account for the difference in intensity value representation and the spatial cooridinates. The aggregation in such a case would take the form

$$\left\{ \begin{array}{cc} Wb_{s}\\ b_{g}\end{array}\right\} .$$ (5)

but this is a practical consideration that need not be a concern at this point.