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diff --git a/report/paper.md b/report/paper.md index c84530e..d286b08 100755 --- a/report/paper.md +++ b/report/paper.md @@ -12,7 +12,9 @@ for training as a standard. This will allow to give more than one example of success and failure for each class when classifying the test_data. - +\begin{center} +\includegraphics[width=20em]{fig/partition.pdf} +\end{center} After partitioning the data into training and testing sets, PCA is applied. The covariance matrix, S, of dimension @@ -23,15 +25,19 @@ training samples minus one. This can be observed in the graph below as a sudden drop for eigenvalues after the 415th. - +\begin{center} +\includegraphics[width=20em]{fig/eigenvalues.pdf} +\end{center} The mean image is calculated averaging the features of the training data. Changing the randomization seed will give very similar values, since the vast majority of the training faces used for averaging will be the same. The mean face for our standard seed can be observed below. - -{ width=1em } + +\begin{center} +\includegraphics[width=8em]{fig/mean_face.pdf} +\end{center} To perform face recognition we choose the best M eigenvectors associated with the largest eigenvalues. We tried @@ -40,7 +46,9 @@ M=42 with accuracy=66.3%. After such value the accuracy starts to flaten, with some exceptions for points at which accuracy decreases. WE NEED TO ADD PHYSICAL MEANINGS - +\begin{center} +\includegraphics[width=20em]{fig/accuracy.pdf} +\end{center} # Question 1, Application of eigenfaces @@ -86,9 +94,9 @@ we obtain a very large matrix. The computation process can get very expensive wh For such reason we compute the eigenvectors **v\textsubscript{i}** of the NxN matrix A\textsuperscript{T}A. From the computation it follows that $$ A\textsuperscript{T}A\boldsymbol{v\textsubscript{i}} = \lambda \textsubscript{i}\boldsymbol{v\textsubscript{i}} $$ -Multiplying both side by A we obtain: $$ AA\textsuperscript{T}A\boldsymbol{v\textsubscript{i}} = \lambda \textsubscript{i}A\boldsymbol{v\textsubscript{i}} $$ +Multiplying both side by A we obtain: -$$ SA\boldsymbol{v\textsubscript{i}} = \lambda \textsubscript{i}A\boldsymbol{v\textsubscript{i}} $$ +$$ AA\textsuperscript{T}A\boldsymbol{v\textsubscript{i}} = \lambda \textsubscript{i}A\boldsymbol{v\textsubscript{i}} \rightarrow SA\boldsymbol{v\textsubscript{i}} = \lambda \textsubscript{i}A\boldsymbol{v\textsubscript{i}} $$ We know that $$ S\boldsymbol{u\textsubscript{i}} = \lambda \textsubscript{i}\boldsymbol{u\textsubscript{i}} $$ @@ -110,7 +118,9 @@ Observing in fact the variance ratio of the principal components, the contributi they'll have will be very low for values above 100, hence we will require a much higher quantity of components to improve reconstruction quality. - +\begin{center} +\includegraphics[width=20em]{fig/variance.pdf} +\end{center} The analysed classification methods used for face recognition are **Nearest Neighbor** and **alternative method** through reconstruction error. @@ -121,58 +131,124 @@ REFER TO ACCURACY GRAPH 1 FOR NN. MAYBE WE CAN ALSO ADD SAME GRAPH WITH DIFFEREN A confusion matrix showing success and failure cases for Nearest Neighbor classfication can be observed below: - +\begin{center} +\includegraphics[width=20em]{fig/pcacm.pdf} +% +\end{center} An example of failed classification is a test face from class 2, wrongly labeled as class 5: - +\begin{center} +\includegraphics[width=20em]{fig/failure_2_5.pdf} +% +\end{center} The alternative method shows overall a better performance, with peak accuracy of 73% for M=3. The maximum M non zero eigenvectors that can be used will in this case be at most the amount of training samples per class minus one, since the same amount of eigenvectors will be used for each generated class-subspace. - +\begin{center} +\includegraphics[width=20em]{fig/alternative_accuracy.pdf} +% +\end{center} A confusion matrix showing success and failure cases for alternative method classfication can be observed below: - +\begin{center} +\includegraphics[width=20em]{fig/altcm.pdf} +% +\end{center} It can be observed that even with this more accurate classification, there is one instance of mislabel of the same face of class 2 as class 5. An additional classification failure of class 6 labeled as class 7 can be observed below: - +\begin{center} +\includegraphics[width=20em]{fig/failure_6_7.pdf} +% +\end{center} -# Part 2 +# Question 2, Generative and Discriminative Subspace Learning Maximize function J(W) (Fisher's Criterion): + $$ J(W) = \frac{W\textsuperscript{T}S\textsubscript{B}W}{W\textsuperscript{T}S\textsubscript{W}W}\textrm{ or } J(W) = \frac{W\textsuperscript{T}S\textsubscript{B}W}{W\textsuperscript{T}S\textsubscript{t}W}$$ + With S\textsubscript{B} being the scatter matrix between classes, S\textsubscript{W} being the within-class scatter matrix and W being the set of projection vectors. $\mu$ represents the mean vector(???) of each class. + $$ S\textsubscript{B} = \sum\limits_{c}(\mu\textsubscript{c} - \overline{x})(\mu\textsubscript{c} - \overline{x})\textsuperscript{T} $$ -$$ S\textsubscript{W} = \sum\limits_{c}\sum\limits_{i\epsilon c}(x\textsubscript{i} - \mu\textsubscript{c})(x\textsubscript{i} - \mu\textsubscript{c})\textsuperscript{T} $$ +$$ S\textsubscript{W} = \sum\limits_{c}\sum\limits_{i\in c}(x\textsubscript{i} - \mu\textsubscript{c})(x\textsubscript{i} - \mu\textsubscript{c})\textsuperscript{T} $$ + To maximize J(W) we differentiate with respect to W and equate to zero: + $$ \frac{d}{dW}J(W) = \frac{d}{dW}(\frac{W\textsuperscript{T}S\textsubscript{B}W}{W\textsuperscript{T}S\textsubscript{W}W}) = 0 $$ $$ (W\textsuperscript{T}S\textsubscript{W}W)\frac{d(W\textsuperscript{T}S\textsubscript{B}W)}{dW} - (W\textsuperscript{T}S\textsubscript{B}W)\frac{d(W\textsuperscript{T}S\textsubscript{W}W)}{dW} = 0 $$ $$ (W\textsuperscript{T}S\textsubscript{W}W)2S\textsubscript{B}W - (W\textsuperscript{T}S\textsubscript{B}W)2S\textsubscript{W}W = 0 $$ $$ S\textsubscript{B}W - JS\textsubscript{W}W = 0 $$ + Multiplying by the inverse of S\textsubscript{W} we obtain: + $$ S\textsubscript{W}\textsuperscript{-1}S\textsubscript{B}W - JW = 0 $$ + From here it follows: -$$ W\textsuperscript{*} = argmax(\frac{W\textsuperscript{T}S\textsubscript{B}W}{W\textsuperscript{T}S\textsubscript{W}W}) = S\textsubscript{W}\textsuperscript{-1}(\mu\textsubscript{1} - \mu\textsubscript{2}) $$ + +$$ W\textsuperscript{*} = arg\underset{W}max(\frac{W\textsuperscript{T}S\textsubscript{B}W}{W\textsuperscript{T}S\textsubscript{W}W}) = S\textsubscript{W}\textsuperscript{-1}(\mu\textsubscript{1} - \mu\textsubscript{2}) $$ + By isomorphic mapping where P are the eigenvectors generated through PCA: + $$ W = PX $$ + We can substitute for W in the J(W) expression, obtaining: + $$ J(W) = \frac{X\textsuperscript{T}P\textsuperscript{T}S\textsubscript{B}PX}{X\textsuperscript{T}P\textsuperscript{T}S\textsubscript{t}PX} $$ + We can rewrite such expression substituting for: + $$ P\textsuperscript{T}S\textsubscript{B}P = \widetilde{S}\textsubscript{B} \textrm{ and } P\textsuperscript{T}S\textsubscript{t}P = \widetilde{S}\textsubscript{t} $$ $$ J(W) = \widetilde{J}(W) = \frac{X\textsuperscript{T}\widetilde{S}\textsubscript{B}X}{X\textsuperscript{T}\widetilde{S}\textsubscript{t}X} $$ -# Conclusion +$$ \widetilde{S}\textsubscript{B} \textrm{ and } \widetilde{S}\textsubscript{t} $$ + +are respectively semi-positive definite and positive definite. So $$ \widetilde{J}(X) $$ +acts like Fisher's criterion but in PCA transformed space. This method +does not result in any loss of data. + +*Proof:* + +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(???) + +**Theorem:** +*$$ \textrm{For any arbitrary } \varphi \in R\textsuperscript{n}, \varphi +\textrm{ can be denoted by } \varphi = X + \xi, $$ +$$ \textrm{ where, }X \in \phi\textsubscript{t}\textrm{ and } \xi \in +\phi\textsubscript{t}\textsuperscript{perp}\textrm{, and +satisfies }J(\varphi)=J(X). $$* + +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. + +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 +# Question 3, LDA Ensemble for Face Recognition, PCA-LDA Ensemble # References |