Merge branch 'master' of skozl.com:e4-pattern

1 files changed, 10 insertions, 7 deletions

diff --git a/report/paper.md b/report/paper.md
index ecf1e3e..2789923 100755
--- a/report/paper.md
+++ b/report/paper.md
@@ -111,15 +111,16 @@ improve identification accuracy, and consider it an additional baseline.
 \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 rank-list 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
@@ -139,7 +140,9 @@ transformations performed the the ResNet-50 convolution model the features were
 \end{center}
 \end{figure}
 
-While we did not use mahalanobis as a primary distance metric, it is possible to use the Mahalanobis metric, together with the next investigated solution $k
+While we did not use mahalanobis as a primary distance metric, it is possible to use the Mahalanobis metric, together with the next investigated solution involving $k$-reciprocal re-ranking.
+
+# Suggested Improvement
 
 ## $k$-reciprocal Re-ranking Formulation
 
@@ -178,11 +181,11 @@ 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 $k_1$: $R^*(g_i,k_1)$.