Point distribution model

The point distribution model is a model for representing the mean geometry of a shape and some statistical modes of geometric variation inferred from a training set of shapes. The point distribution model is a model for representing the mean geometry of a shape and some statistical modes of geometric variation inferred from a training set of shapes. The point distribution model concept has been developed by Cootes, Taylor et al. and became a standard in computer vision for the statistical study of shape and for segmentation of medical images where shape priors really help interpretation of noisy and low-contrasted pixels/voxels. The latter point leads to active shape models (ASM) and active appearance models (AAM). Point distribution models rely on landmark points. A landmark is an annotating point posed by an anatomist onto a given locus for every shape instance across the training set population. For instance, the same landmark will designate the tip of the index finger in a training set of 2D hands outlines. Principal component analysis (PCA), for instance, is a relevant tool for studying correlations of movement between groups of landmarks among the training set population. Typically, it might detect that all the landmarks located along the same finger move exactly together across the training set examples showing different finger spacing for a flat-posed hands collection. First, a set of training images are manually landmarked with enough corresponding landmarks to sufficiently approximate the geometry of the original shapes. These landmarks are aligned using the generalized procrustes analysis, which minimizes the least squared error between the points. k {displaystyle k} aligned landmarks in two dimensions are given as It's important to note that each landmark i ∈ { 1 , … k } {displaystyle iin lbrace 1,ldots k brace } should represent the same anatomical location. For example, landmark #3, ( x 3 , y 3 ) {displaystyle (x_{3},y_{3})} might represent the tip of the ring finger across all training images. Now the shape outlines are reduced to sequences of k {displaystyle k} landmarks, so that a given training shape is defined as the vector X ∈ R 2 k {displaystyle mathbf {X} in mathbb {R} ^{2k}} . Assuming the scattering is gaussian in this space, PCA is used to compute normalized eigenvectors and eigenvalues of the covariance matrix across all training shapes. The matrix of the top d {displaystyle d} eigenvectors is given as P ∈ R 2 k × d {displaystyle mathbf {P} in mathbb {R} ^{2k imes d}} , and each eigenvector describes a principal mode of variation along the set. Finally, a linear combination of the eigenvectors is used to define a new shape X ′ {displaystyle mathbf {X} '} , mathematically defined as: where X ¯ {displaystyle {overline {mathbf {X} }}} is defined as the mean shape across all training images, and b {displaystyle mathbf {b} } is a vector of scaling values for each principal component. Therefore, by modifying the variable b {displaystyle mathbf {b} } 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 b {displaystyle mathbf {b} } to be within ± {displaystyle pm } 3 standard deviations, where the standard deviation of a given principal component is defined as the square root of its corresponding eigenvalue.