Article: abstract and width control revisited

Article/abstract.tex

-\begin{abstract}
-  Penser \`a prendre du beurre sal\'e \`a la biocoop.
-\end{abstract}
+  This paper introduces a new straight edge detector in gray-level images
+based on blurred segments, digital objects able to imbed quality measurements
+on the extracted features. This study completes previous works with a better
+estimation of the blurred segment width and orientation, with the help of two
+main improvements : adaptive directional scans and the control of the
+assigned width to the recognition algorithm.

Article/main.tex

@@ -34,9 +34,8 @@
     \maketitle
 
     \begin{abstract}
-TOWRITE.
-    
-    \keywords{Line detection \and discrete geometry \and TOCOMPLETE.}
+        \input{abstract}  
+    \keywords{Line detection \and discrete geometry \and ONE MORE PLEASE.}
     \end{abstract}
 
   \end{frontmatter}

Article/method.tex

@@ -52,15 +52,10 @@ The fine tracking step consists on building and extending a blurred segment
 $\mathcal{B}_2$ based on points that correspond to local maxima of the
 image gradient, ranked by magnitude order, and with gradient direction
 close to a reference gradient direction at the segment first point.
-This step uses an adaptive directional scanner based on the found
-position $C$ direction $\vec{D}$ in order to extends the segment in the
+At this refinement step, the control of the assigned width is applied
+and an adaptive directional scanner based on the found position $C$ and
+direction $\vec{D}$ is used in order to extends the segment in the
 appropriate direction.
-After $N$ points are added without any augmentation of the segment minimal
-width, this width becomes the new assigned width so that the segment
-can not thicken any more. This procedure allows to control the blurred
-segment width based on the observation of its evolution in the vicinity
-of the input stroke.
-Setting $N=20$ shows a good behaviour on tested images.
 
 The fine track output segment is finally filtered to remove artifacts
 and outliers, and a solution blurred segment $\mathcal{B}_3$ is provided.

Article/notions.tex

@@ -52,15 +52,6 @@ the assigned width $\varepsilon$, then the new input point is rejected.}
   \label{fig:bs}
 \end{figure}
 
-The control of the assigned width $\varepsilon$ is ensured on the
-following way.
-At the beginning, a large width $\varepsilon_0$ is assigned to the
-recognition problem to allow the detection of large blurred segments.
-Then, when no more aumentation of the minimal width is observed as the
-segment grows ($\mu_{i+\lambda} = \mu_i$), the assigned width is set at
-a near value to the observed minimal width in order to avoid the
-incorporation of spurious outliers in further parts of the segment.
-
 \subsection{Directional scan}
 
 A directional scan $DS$ is an ordered partition restricted to the image
@@ -214,8 +205,8 @@ An example of adaptive directional scan is given in \RefFig{fig:adaption}.
 \begin{figure}[h]
 \center
   \begin{tabular}{c@{\hspace{0.2cm}}c}
-    \includegraphics[width=0.49\textwidth]{Fig_notions/adaptionBounds_zoom.png} &
-    \includegraphics[width=0.49\textwidth]{Fig_notions/adaptionLines_zoom.png}
+    \includegraphics[width=0.49\textwidth]{Fig_notions/adaptionBounds_zoom.png}
+    & \includegraphics[width=0.49\textwidth]{Fig_notions/adaptionLines_zoom.png}
   \end{tabular}
   \caption{Example of blurred segment detection
            using an adaptive directional scan.
@@ -225,3 +216,20 @@ An example of adaptive directional scan is given in \RefFig{fig:adaption}.
            Adaption is quite sensible when crossing the tile joins.}
   \label{fig:adaption}
 \end{figure}
+
+\subsection{Control of the assigned width}
+
+The assigned width $\varepsilon$ to the blurred segment recognition algorithm
+is initially set to a large value $\varepsilon_0$ in order to allow the
+detection of large blurred segments.
+Then, when no more augmentation of the minimal width is observed as the
+blurred segment expends ($\mu_{i+\lambda} = \mu_i$), it is set to a much
+stricter value able to circumscribe the possible interpretations of the
+segment, that take into account the digitization margins:
+\begin{equation}
+\varepsilon = \mu_{i+\lambda} + 1/2
+\end{equation}
+This strategy aims at preventing the incorporation of spurious outliers in
+further parts of the segment.
+Setting the observation distance to a constant value $\lambda = 20$ seems
+appropriate in most experimented situations.