Complete rerank intro

1 files changed, 17 insertions, 2 deletions

diff --git a/report2/paper.md b/report2/paper.md
index 05f0ac1..8e76142 100755
--- a/report2/paper.md
+++ b/report2/paper.md
@@ -108,7 +108,7 @@ We find that for the query and gallery set clustering does not seem to improve i
 
 # Suggested Improvement
 
-## k-reciprocal Reranking
+## k-reciprocal Reranking Formulation
 
 The approach addressed to improve the identification performance is based on
 k-reciprocal reranking. The following section summarizes the idea behind
@@ -120,7 +120,7 @@ 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
+the problem of 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.
 
@@ -147,6 +147,21 @@ through Jaccardian metric as:
 
 $$ d_J(p,g_i)=1-\frac{\sum\limits_{j=1}^N min(V_{p,g_j},V_{g_i,g_j})}{\sum\limits_{j=1}^N max(V_{p,g_j},V_{g_i,g_j})} $$
 
+It is then possible to perform a local query expansion using the g\textsubscript{i} neighbors of
+defined as $V_p=\frac{1}{|N(p,k_2)|}\sum\limits_{g_i\in N(p,k_2)}V_{g_i}$. 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 $k_1$:$R^*(g_i,k_1)$.
+
+The distances obtained are then mixed, obtaining a final distance $d^*(p,g_i)$ that is used to obtain the
+improved ranklist: $d^*(p,g_i)=(1-\lambda)d_J(p,g_i)+\lambda d(p,g_i)$.
+
+The aim is to learn optimal values for $k_1,k_2$ and $\lambda$ in the training set that improve top1 identification accuracy.
+This is done through a simple **GRADIENT DESCENT** algorithm followed by exhaustive search to estimate 
+$k_{1_{opt}}$ and $k_{2_{opt}}$ for eleven values of $\lambda$ from zero(only Jaccard distance) to one(only original distance)
+in steps of 0.1. The results obtained through this approach suggest: $k_{1_{opt}}=9, k_{2_{opt}}=3, 0.1\leq\lambda_{opt}\leq 0.3$.
+
+## k-reciprocal Reranking Formulation
+
 \begin{figure}
 \begin{center}
 \includegraphics[width=24em]{fig/ranklist.png}