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m →Background: changed index to "index finger" for clarity, since that is what's being discussed. |
m →Details: verb tense |
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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 compute normalized [[eigenvectors]] and [[eigenvalues]] of the [[covariance matrix]] across all training shapes. The matrix of the top <math>d</math> eigenvectors is given as <math>\mathbf{P} \in \mathbb{R}^{2k \times d}</math>, and each eigenvector
Finally, a [[linear combination]] of the eigenvectors is used to define a new shape <math>\mathbf{X}'</math>, mathematically defined as:
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