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diff --git a/report/paper.md b/report/paper.md index 6291273..8dc0421 100755 --- a/report/paper.md +++ b/report/paper.md @@ -12,15 +12,8 @@ accuracy is obtained when using a 90% of the data for training. Despite such results we will be using 70% of the data 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{figure} -\begin{center} -\includegraphics[width=20em]{fig/partition.pdf} -\label{accuracy} -\caption{NN Recognition Accuracies for different data partitions} -\end{center} -\end{figure} +test_data. Moreover using 90% training data would make the results +obtained heavilly dependent on the seed chosen. After partitioning the data into training and testing sets, PCA is applied. The covariance matrix, S, of dimension @@ -118,7 +111,7 @@ From here it follows that AA\textsuperscript{T} and A\textsuperscript{T}A have t It can be noticed that we effectively don't lose any data calculating the eigenvectors for PCA with the second method. The main advantages of it are in terms of speed, -(since the two methods require on average respectively 3.4s and 0.14s), and complexity of computation +(since the two methods require on average respectively 3.4s and 0.11s), and complexity of computation (since the eigenvectors found with the first method are extracted from a significantly bigger matrix). @@ -131,7 +124,7 @@ the covariance matrix, whereas method 2 requires an additional projection step. Using the computational method for fast PCA, face reconstruction is then performed. The quality of reconstruction will depend on the amount of eigenvectors picked. -The results of varying M can be observed in the picture in fig.\ref{face160rec}. Two faces from classes +The results of varying M can be observed in fig.\ref{face160rec}. Two faces from classes number 21 and 2 respectively, are reconstructed as shown in fig.\ref{face10rec} with respective M values of M=10, M=100, M=200, M=300. The last picture is the original face. @@ -198,7 +191,7 @@ classification. It is possible to use a NN classification that takes into account majority voting. With this method recognition is based on the K closest neighbors of the projected test image. Such method anyways showed the best recognition accuracies for PCA with -K=1, as it can be observed from the figure \ref{k-diff}. +K=1, as it can be observed from figure \ref{k-diff}. \begin{figure} \begin{center} @@ -237,7 +230,6 @@ can be observed in figure \ref{cm-alt}. \end{figure} Similarly to the NN case, we present two cases, respectively failure and success. -The pictures on the right show the reconstructed images. \begin{figure} \begin{center} @@ -281,7 +273,7 @@ $$ S\textsubscript{W} = \sum\limits_{c}\sum\limits_{i\in c}(x\textsubscript{i} - To maximize J(W) we differentiate with respect to W and equate to zero: -$$ \frac{d}{dW}J(W) = \frac{d}{dW}(\frac{W\textsuperscript{T}S\textsubscript{B}W}{W\textsuperscript{T}S\textsubscript{W}W}) = 0 $$ +$$ \frac{d}{dW}J(W) = \frac{d}{dW}\left(\frac{W\textsuperscript{T}S\textsubscript{B}W}{W\textsuperscript{T}S\textsubscript{W}W}\right) = 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 $$ @@ -291,15 +283,15 @@ $$ S\textsubscript{W}\textsuperscript{-1}S\textsubscript{B}W - JW = 0 $$ From here it follows: -$$ W\textsubscript{opt} = 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}) $$ +$$ W\textsubscript{opt} = 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}) $$ However S\textsubscript{W} is often singular since the rank of S\textsubscript{W} is at most N-c and usually N is smaller than D. In such case it is possible to use Fisherfaces. The optimal solution to such -problem lays in W\textsuperscript{T}\textsubscript{opt} = W\textsuperscript{T}\textsubscript{lda}W\textsuperscript{T}\textsubscript{pca} +problem lays in W\textsuperscript{T}\textsubscript{opt} = W\textsuperscript{T}\textsubscript{lda}W\textsuperscript{T}\textsubscript{pca}, -Where W\textsubscript{pca} is chosen to maximize the determinant of the total scatter matrix +where W\textsubscript{pca} is chosen to maximize the determinant of the total scatter matrix of the projected samples: $$ W\textsuperscript{T}\textsubscript{pca} = arg\underset{W}max|W\textsuperscript{T}S\textsubscript{T}W| $$ $$ And $$ $$ W\textsubscript{lda} = arg\underset{W}max\frac{|W\textsuperscript{T}W\textsuperscript{T}\textsubscript{pca}S\textsubscript{B}W\textsubscript{pca}W|}{|W\textsuperscript{T}W\textsuperscript{T}\textsubscript{pca}S\textsubscript{W}W\textsubscript{pca}W|} $$ @@ -332,7 +324,7 @@ vaying between 0.11s(low M_pca) and 0.19s(high M_pca). \end{center} \end{figure} -DD RANK OF SCATTER MATRICES +ADD RANK OF SCATTER MATRICES Testing with M_lda=50 and M_pca=115 gives 92.9% accuracy. The results of such test can be observed in the confusion matrix shown in figure \ref{ldapca_cm}. |