From adf19301ef6e661d09351f23b939605f229c10d2 Mon Sep 17 00:00:00 2001 From: nunzip Date: Wed, 7 Nov 2018 16:32:37 +0000 Subject: Version 1 --- report/paper.md | 109 ++++++++++++++++++++++++++++++++++++++------------------ 1 file changed, 75 insertions(+), 34 deletions(-) mode change 100644 => 100755 report/paper.md diff --git a/report/paper.md b/report/paper.md old mode 100644 new mode 100755 index 8ea9791..113dfa6 --- a/report/paper.md +++ b/report/paper.md @@ -1,49 +1,90 @@ -# Introduction +# Question 1, Eigenfaces -An introduction. Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed -do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim -veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo -consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse -cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non -proident, sunt in culpa qui officia deserunt mollit anim id est laborum. +The data is partitioned to allow random selection of the +same amount of samples for each class. This is done to +prevent overfitting (?) of some classes with respect to others. In +such way, each training vector space will be generated with +the same amount of elements. The test data will instead +be taken from the remaining samples. Testing on accuracy +with respect to data partition indicates that the maximum +accuracy is obtained when using a 90% of the data for +training. Despite such results we will be using 80% 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. -# Method +![Classification Accuracy of Test Data vs % of data used for training](fig/partition.pdf "Partition") -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod -tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At -vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, -no sea takimata sanctus est Lorem ipsum dolor sit amet. +After partitioning the data into training and testing sets, +PCA is applied. The covariance matrix, S, of dimension +2576x2576 (features x features), will have 2576 eigenvalues +and eigenvectors. The amount of non-zero eigenvalues and +eigenvectors obtained will only be equal to the amount of +training samples minus one. This can be observed in the +graph below as a sudden drop for eigenvalues after the +415th. -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod -tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At -vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, -no sea takimata sanctus est Lorem ipsum dolor sit amet. +![Log PCA Eigenvalues](fig/eigenvalues.pdf "Eigenvalues") -# Footnotes +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. -Example of footnote^[A footnote example]. Lorem ipsum dolor sit amet, consectetur -adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna -aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi -ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in -voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint -occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim -id est laborum. +![Mean Face](fig/mean_face.pdf){ width=1em } -# Cites -Zotero + Better BibTex. All cites are on the file bibliography.bib. This is -a cite[@djangoproject_models_2016]. +To perform face recognition we choose the best M eigenvectors +associated with the largest eigenvalues. We tried +different values of M, and we found an optimal point for +M=120. After such value the accuracy starts to flaten, with +some exception for points at which accuracy decreases. +WE NEED TO ADD PHYSICAL MEANINGS -# Conclusion +![Recognition Accuracy of Test data varying M](fig/accuracy.pdf "Accuracy1") + +# Question 1, Application of eigenfaces + +rming the low-dimensional computation of the +eigenspace for PCA we obtain the same accuracy results +of the high-dimensional computation previously used. A +comparison between eigenvalues and eigenvectors of the +two computation techniques used shows that the difference +is very small. The difference we observed is due to rounding +of the np.eigh function when calculating the eigenvalues +and eigenvectors of the matrices ATA (DxD) and AAT +(NxN). + +The first ten biggest eigenvalues obtained with each method +are shown in the table below. -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod -tempor invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At -vero eos et accusam et justo duo dolores et ea rebum. Stet clita kasd gubergren, -no sea takimata sanctus est Lorem ipsum dolor sit amet. +\begin{table}[ht] +\centering +\begin{tabular}[t]{cc} +PCA &Fast PCA\\ +2.9755E+05 &2.9828E+05\\ +1.4873E+05 &1.4856E+05\\ +1.2286E+05 &1.2259E+05\\ +7.5084E+04 &7.4950E+04\\ +6.2575E+04 &6.2428E+04\\ +4.7024E+04 &4.6921E+04\\ +3.7118E+04 &3.7030E+04\\ +3.2101E+04 &3.2046E+04\\ +2.7871E+04 &2.7814E+04\\ +2.4396E+04 &2.4339E+04\\ +\end{tabular} +\caption{Comparison of eigenvalues obtain with the two computation methods} +\end{table} -If you want to write an equation: +It can be proven that the eigenvalues and eigenvectors +obtain are the same: ##PROVE -$$ x^2 = \frac{\pi}{2} $$ +Reconstruction is then performed on a chosen +# Cites + + +# Conclusion # References -- cgit v1.2.3-54-g00ecf