Minor layout adjustments

1 files changed, 11 insertions, 11 deletions

diff --git a/report/paper.md b/report/paper.md
index 79d4183..79612c3 100755
--- a/report/paper.md
+++ b/report/paper.md
@@ -16,7 +16,7 @@ The dataset CUHK03 contains 14096 pictures of people captured from two
 different cameras. The feature vectors used, extracted from a trained ResNet50 model
 , contain 2048 features that are used for identification.
 
-The pictures represent 1467 different identities, each of which appears 9 to 10
+The pictures represent 1467 different identities, each of which appears 7 to 10
 times. Data is seperated in train, query and gallery sets with `train_idx`, 
 `query_idx` and `gallery_idx` respectively, where the training set has been used 
 to develop the ResNet50 model used for feature extraction. This procedure has 
@@ -41,9 +41,6 @@ distance:
 
 $$ \textrm{NN}(x) = \operatorname*{argmin}_{i\in[m]} \|x-x_i\| $$
 
-Alternative distance metrics exist such as jaccardian and mahalanobis, which can
-be used as an alternative to euclidiean distance.
-
 # Baseline Evaluation
 
 To evaluate improvements brought by alternative distance learning metrics a baseline 
@@ -105,21 +102,22 @@ We find that for the query and gallery set clustering does not seem to improve i
 \end{center}
 \end{figure}
 
-# Suggested Improvement
-
 ## Mahalanobis Distance
 
 We were not able to achieve significant improvements using mahalanobis for 
 original distance ranking compared to square euclidiaen metrics. 
 
-The mahalanobis distance metric was used to create the ranklist as an alternative to euclidean distance.
-When performing mahalanobis with the training set as the covariance matrix, reported accuracy is reduced to **38%** .
+The mahalanobis distance metric was used to create the ranklist as an alternative to euclidean distance:
+
+$$ d_M(p,g_i) = (p-g_i)^TM(p-g_i). $$
+
+When performing mahalanobis with the covariance matrix $M$ generated from the training set, reported accuracy is reduced to **38%** .
 
 We also attempted to perform the same mahalanobis metric on a reduced PCA featureset. This allowed for significant execution 
 time improvements due to the greatly reduced computation requierments for smaller featurespace, but nevertheless demonstrated no
 improvements over an euclidean metric.
 
-These results are likely due to the **extremely** low covariance of features in the training set. This is evident when looking at the Covariance matrix of the training data, and is also visible in figure \ref{fig:subspace}. This is likely the result of the feature transformations performed the the ResNet-50 convolution model the features were extracted from.
+These results are likely due to the **extremely** low covariance of features in the training set. This is evident when looking at the covariance matrix of the training data, and is also visible in figure \ref{fig:subspace}. This is likely the result of the feature transformations performed the the ResNet-50 convolution model the features were extracted from.
 
 \begin{figure}
 \begin{center}
@@ -130,6 +128,8 @@ These results are likely due to the **extremely** low covariance of features in
 \end{center}
 \end{figure}
 
+# Suggested Improvement
+
 ## $k$-reciprocal Re-ranking Formulation
 
 The approach addressed to improve the identification performance is based on
@@ -167,11 +167,11 @@ e\textsuperscript{\textit{-d(p,g\textsubscript{i})}}, & \text{if}\ \textit{g\tex
 Through this transformation it is possible to reformulate the distance obtained 
 through Jaccardian metric as: 
 
-$$ d_J(p,g_i)=1-\frac{\sum\limits_{j=1}^N min(V_{p,g_j},V_{g_i,g_j})}{\sum\limits_{j=1}^N max(V_{p,g_j},V_{g_i,g_j})} $$
+$$ d_J(p,g_i)=1-\frac{\sum\limits_{j=1}^N min(V_{p,g_j},V_{g_i,g_j})}{\sum\limits_{j=1}^N max(V_{p,g_j},V_{g_i,g_j})}. $$
 
 It is then possible to perform a local query expansion using the g\textsubscript{i} neighbors of
 defined as:
-$$ V_p=\frac{1}{|N(p,k_2)|}\sum\limits_{g_i\in N(p,k_2)}V_{g_i} $$. 
+$$ V_p=\frac{1}{|N(p,k_2)|}\sum\limits_{g_i\in N(p,k_2)}V_{g_i}. $$ 
 We refer to $k_2$ since we limit the size of the nighbors to prevent noise 
 from the $k_2$ neighbors. The dimension k of the *$R^*$* set will instead 
 be defined as $k_1$: $R^*(g_i,k_1)$.