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Diffstat (limited to 'report')
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diff --git a/report/paper.md b/report/paper.md index de62e4f..9cb14c7 100755 --- a/report/paper.md +++ b/report/paper.md @@ -229,6 +229,8 @@ The pictures on the right show the reconstructed images. \end{center} \end{figure} +From the failures and success cases analyzed it is noticeable that the parameters that +affect recognition the most are: glasses, hair, sex and brightness of the picture. # Question 2, Generative and Discriminative Subspace Learning @@ -282,20 +284,13 @@ are respectively semi-positive definite and positive definite. $\widetilde{J}(X) similarly to the original J(X), applies Fisher's criterion in a PCA generated subspace. This enables to perform LDA minimizing loss of data. -*Proof:* +R\textsuperscript{n} contains the optimal discriminant vectors for LDA. +However S\textsubscript{t} is singular and the vectors are found through +an expensive computational process. The soultion to such issue is derivation +from a lower space. -REWRITE FROM HERE - -The set of optimal discriminant vectors can be found in R\textsuperscript{n} -for LDA. But, this is a difficult computation because the dimension is very -high. Besides, S\textsubscript{t} is always singular. Fortunately, it is possible -to derive it to a lower dimensional space. - -Suppose **b\textsubscript{n}** are the eigenvectors of S\textsubscript{t}. -The M biggest eigenvectors are positive: M = *rank*(S\textsubscript{t}). -The other M+1 to compose the null space of S\textsubscript{t}. - -YO WTF(???) +The M biggest eigenvectors of S\textsubscript{t} are positive and non zero and +*rank*(S\textsubscript{t})=M. **Theorem:** *$$ \textrm{For any arbitrary } \varphi \in R\textsuperscript{n}, \varphi @@ -304,14 +299,14 @@ $$ \textrm{ where, }X \in \phi\textsubscript{t}\textrm{ and } \xi \in \phi\textsubscript{t}\textsuperscript{perp}\textrm{, and satisfies }J(\varphi)=J(X). $$* -According to the theorem, it is possible to find optimal discriminant -vectors, reducing the problem dimension without any loss of information -with respect to Fisher’s criterion. +The theorem indicates that the optimal discriminant vectors can be derived +through the reduced space obtained through PCA without losing information +according to the Fisher's criterion. +In conclusion such method is theoretically better than LDA and PCA alone. +The Fisherfaces method requires less computation complexity and less time than +LDA and it improves recognition performances with respect to PCA and LDA. Fisherfaces method is effective because it requires less computation -time than regular LDA. It also has lower error rates compared to -Eigenfaces method. Thus, it combines the performances of discriminative -and reconstructive tools. # Question 3, LDA Ensemble for Face Recognition, PCA-LDA |