Point distribution model: Difference between revisions

Revision as of 20:00, 28 December 2009

Revision as of 20:04, 28 December 2009

Line 36:

It's important to note that each landmark <math>i \in \lbrace 1, \ldots k \rbrace </math> should represent the same anatomical ___location. For example, landmark #3, <math>(x_3, y_3)</math> might represent the tip of the ring finger across all training images.

Now the shape outlines are reduced to sequences of <math>k</math> landmarks, so that a given training shape is defined as the vector <math>\mathbf{X} \in \mathbb{R}^{2k}</math>. Assuming the scattering is [[gaussian distribution|gaussian]] in this space, PCA is used to computes normalized [[eigenvectors]] and [[eigenvalues]] of the [[covariance matrix]] across all training shapes. The matrix of the top few<math>d</math> eigenvectors is given as <math>\mathbf{P} \in \mathbb{R}^{2k \times d}</math>, and each eigenvector describe a principal mode of variation along the set.

Finally, a [[linear_combination|linear combination]] of the eigenvectors is used to define a new shape <math>\mathbf{X}'</math>, mathematically defined as:

Line 42:

:<math>\mathbf{X}' = \overline{\mathbf{X}} + \mathbf{P} * \mathbf{b}</math>

where <math>\overline{\mathbf{X}}</math> is defined as the mean shape across all training images, and <math>\mathbf{b}</math> is a vector of scaling vectors for each principal component. It'sTherefore, by modifying the variable <math>\mathbf{b}</math> an infinite number of shapes can be defined. To ensure that the new shapes are all within the variation seen in the training set, it is common to only allow each element of <math>\mathbf{b}</math> to be within <math>\pm</math>3 standard deviations, where the standard deviation of a given principal component is defined as the square root of its corresponding eigenvalue.

PDM's can be extended to any arbitrary number of dimensions, but are typically used in 2D image and 3D volume applications (where each landmark point is <math>\mathbb{R}^2</math> or <math>\mathbb{R}^3</math>).