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Shape Model

To begin encoding the form of an object, landmarks need to be identified and statistical analysis applied so that it expresses these spatial shape properties, namely the landmark coordinates. From this analysis, a mean shape is obtained and it can be denoted by $\mathbf{x}_{mean}$ or $\mathbf{\overline{x}}$. To obtain this mean, the procedure that is commonly used is Procrustes analysis. The generalised Procrustes procedure (or GPA for Generalised Procrustes Analysis) was developed by Gower in 1975 and has been adapted for shape analysis by Goodall in 1991. It processes each component of the vectors derived from the images and returns for each component a value that is said to be the mean. From here onwards, this vector which represents the mean of the data will be referred to as $\mathbf{\overline{x}}$. Each shape $\mathbf{x}$ is then well-formulated by the following:

$$\mathbf{x}=\mathbf{\overline{x}}+\mathbf{P}_{s}\mathbf{b}_{s}.$$ (2.2)

The matrix $\mathbf{P}$ represents the Eigen-vectors of the covariance matrix (set of orthogonal modes of variation) and the parameters $\mathbf{b}_{s}$ control the variation of the shape by altering these modes of variation. The parameters essentially describe the magnitude of the covariance of each element in the matrix. These parameters and the range within which they must lie describe a level of freedom - that is - the freedom (or otherwise constraints) of the model.

Eigen-analysis is used quite extensively in the derivation of the expression above^2.12, but it will not be discussed in detail in the remainder of this report. Instead, a short explanation will be given on Principal Component^2.13 Analysis [,] which from here onwards be referred to as PCA. What is worth emphasising is that the only variant in the model described above is $\mathbf{b}_{s}$ and as the values of $\mathbf{b}_{1<i<s}$ are infinite ( $\mathbf{b}_{1<i<s}$ $\in\mathbb{\mathbb{Z}}$), the same must hold for $\mathbf{x}$. There is an infinite number of shapes, each of which can be generated from one choice of value for each model parameter. One interesting alternative to PCA was presented in [] By Jebara. It is explained at the end of Appendix on page .