Changes between Version 1 and Version 2 of recipes/PCA/explanation


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Timestamp:
Mar 29, 2012, 5:39:43 PM (13 years ago)
Author:
Dave Hunter
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  • recipes/PCA/explanation

    v1 v2  
    11'''Mathematical Defintion''':
    2 {{{
    3 PCA is similar to Factor Analysis, in that will we decompose your data into a set of components that maximise variance in your data.
    4 }}}
     2> PCA is similar to Factor Analysis, in that will we decompose your data into a set of components that maximise variance in your data.
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    65Ok, let's take a step back. It's simple to make an average of a set templates, for each point in the template you take the sum of the x coordinates (horisontal) and devide by the number of templates, you do the same for the y coordinates (vertical). The resulting point is roughly in the middle of the original positions in all the templates. Its exact position is called the centre of gravity, in the it is biased to locations with high point density. So, now you have an average, and it still looks like a face, if it doesn't you're doing it wrong and should look at [wiki:averaging this tutorial].
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     7[[Image(average.jpg)]][[BR]]Example of an average template.
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    89It is possible to describe a face in your dataset as the amount by which that face deviates from the average. For each point in the face's template we subtract (separately, in both x and y coordinates) the coordinates of the corresponding point in the average template. We now have the deviation of each point from the average, this is useful as we could, for example, work out how 'unusual' a face is. The more a face deviates from the average, the more 'unusual' it is. These measure have been shown to be linked to perceptions of attractiveness and familiarity (amongst many others).
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    1415To recap, a face in a PCA model is described as a weighted sum of the PCA components, plus the average face template.
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     17||Example of face shape PCA using Psychomorph templates||
     18||[[Image(average.jpg,width=100%)]]||+||[[Image(one.jpg,width=100%)]]||+||[[Image(two.jpg,width=100%)]]||+||[[Image(three.jpg,width=100%)]]||+||[[Image(four.jpg,width=100%)]]||
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     20||Average||||First Component||||Second Component||||Third Component||||Fourth Component||
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    1623PCA models can be built using Psychomorph’s ASM and PCI functions. ASM (Active Shape Model) applies PCA to templates as explained above. PCI (Principal Component Images) applies PCA to the pixels of an image set.