From 7c162e00d20e031b778c8cf1b6ddef1b4052c8f8 Mon Sep 17 00:00:00 2001
From: nunzip <np.scarh@gmail.com>
Date: Wed, 12 Dec 2018 00:54:58 +0000
Subject: Start Jaccard

---
 report2/paper.md | 47 +++++++++++++++++++++++++++++++++++++++++------
 1 file changed, 41 insertions(+), 6 deletions(-)

(limited to 'report2/paper.md')

diff --git a/report2/paper.md b/report2/paper.md
index 68b803a..05f0ac1 100755
--- a/report2/paper.md
+++ b/report2/paper.md
@@ -40,10 +40,7 @@ of its nearest neighbor(s). The distance between images can be calculated throug
 different distance metrics, however one of the most commonly used is euclidean
 distance:
 
-$$ \textrm{NN}(x) = \operatorname*{argmin}_{i\in[m]} \|x-x_i\|^2 $$
-
-*Square root when calculating euclidean distance is ommited as it does not 
-affect ranking by 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.
@@ -113,6 +110,43 @@ We find that for the query and gallery set clustering does not seem to improve i
 
 ## k-reciprocal Reranking
 
+The approach addressed to improve the identification performance is based on
+k-reciprocal reranking. The following section summarizes the idea behind
+the method illustrated in **REFERENCE PAPER**.
+
+We define $N(p,k)$ as the top k elements of the ranklist generated through NN,
+where p is a query image. The k reciprocal ranklist, $R(p,k)$ is defined as the
+intersection $R(p,k)=\{g_i|(g_i \in N(p,k))\land(p \in N(g_i,k))\}$. Adding 
+$\frac{1}{2}k$ reciprocal nearest neighbors of each element in the ranklist
+$R(p,k)$, it is possible to form a more reliable set 
+$R^*(p,k) \longleftarrow R(p,k) \cup R(q,\frac{1}{2}k)$ that aims to overcome 
+the problem query and gallery images being affected by factors such
+as position, illumination and foreign objects. $R^*(p,k)$ is used to 
+recalculate the distance between query and gallery images.
+
+Jaccard metric of the k-reciprocal sets is used to calculate the distance 
+between p and $g_i$ as: $$d_J(p,g_i)=1-\frac{|R^*(p,k)\cap R^*(g_i,k)|}{|R^*(p,k)\cup R^*(g_i,k)|}$$.
+
+However, since the neighbors of the query p are close to $g_i$ as well, 
+they would be more likely to be identified as true positive. This implies
+the need of a more discriminative method, which is achieved
+encoding the k-reciprocal neighbors into an N-dimensional vector as a function
+of the original distance (in our case square euclidean $d(p,g_i) = \|p-g_i\|^2$) 
+through the gaussian kernell:
+
+\begin{equation}
+\textit{V\textsubscript{p,g\textsubscript{i}}}=
+\begin{cases}
+e\textsuperscript{\textit{-d(p,g\textsubscript{i})}}, & \text{if}\ \textit{g\textsubscript{i}}\in \textit{R\textsuperscript{*}(p,k)} \\
+0, & \text{otherwise.}
+\end{cases}
+\end{equation}
+
+Through this transformation it is possible to reformulate the distance obtained 
+through Jaccardian metric as: 
+
+$$ d_J(p,g_i)=1-\frac{\sum\limits_{j=1}^N min(V_{p,g_j},V_{g_i,g_j})}{\sum\limits_{j=1}^N max(V_{p,g_j},V_{g_i,g_j})} $$
+
 \begin{figure}
 \begin{center}
 \includegraphics[width=24em]{fig/ranklist.png}
@@ -152,8 +186,9 @@ We find that for the query and gallery set clustering does not seem to improve i
 We were not able to achieve significant improvements using mahalanobis for 
 original distance ranking compared to square euclidiaen metrics. Results can 
 be observed using the `-m|--mahalanobis` when running evalution with the 
-repository complimenting this paper. It has been verified that the inverse
-covariance matrix used is PSD, satisfying $d_a(x_1,x_2) \geq 0$.
+repository complimenting this paper.
+
+COMMENT ON VARIANCE AND MAHALANOBIS RESULTS
 
 # Conclusion
 
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