Article: ADS def generalized

Article/biblio.bib

@@ -11,7 +11,7 @@
   title = {Blurred segments in gray level images for
            interactive line extraction},
   author = {Kerautret, Bertrand and Even, Philippe},
-  booktitle = {Proc. of Int. Workshop on Computer Image Analysis}},
+  booktitle = {Proc. of Int. Workshop on Combinatorial Image Analysis},
   series = {LNCS},
   volume = {5852},
   optpublisher = {Springer},

Article/method.tex

@@ -116,9 +116,9 @@ thus producing a useless computational cost.
 
 Here the proposed solution is to dynamically align the scan direction to
 the blurred segment one all along the expansion stage.
-At each iteration $i$, the scan strip is updated using the direction
-of the blurred segment computed at previous iteration $i-1$.
-The adaptive directional scan $ADS$ is then defined by :
+At each iteration $i$, the scan strip is aligned on the direction of the
+blurred segment $\mathcal{B}_{i-1}$ computed at previous iteration $i-1$.
+More generally, an adaptive directional scan $ADS$ is defined by:
 \begin{equation}
 %S_i = \mathcal{D}_{i-1} \cap \mathcal{N}_i
 ADS = \left\{
@@ -126,14 +126,23 @@ S_i = \mathcal{D}_i \cap \mathcal{N}_i \cap \mathcal{I}
 \left| \begin{array}{l}
 \delta(\mathcal{N}_i) = - \delta^{-1}(\mathcal{D}_0) \\
 \wedge~ h_0(\mathcal{N}_i) = h_0(\mathcal{N}_{i-1}) + p(\mathcal{D}) \\
-\wedge~ \mathcal{D}_{i} = D (\mathcal{B}_{i-1},\varepsilon + k), i > 1
+\wedge~ \mathcal{D}_{i} = \mathcal{D} (C_{i-1}, \vec{D}_{i-1}, w_{i-1}), i > 1
+%\wedge~ \mathcal{D}_{i} = D (\mathcal{B}_{i-1},\varepsilon + k), i > 1
 \end{array} \right. \right\}
 \end{equation}
-where $D (\mathcal{B}_i,w)$ is the scan strip aligned to the
-detected segment at iteration $i$ with width $w$.
-In practice, the scan width is set a little greater than the assigned
-width $\varepsilon$ ($k$ is a constant arbitrarily set to 4).
-The last clause expresses the update of the scan bounds at iteration $i$.
+%where $D (\mathcal{B}_i,w)$ is the scan strip aligned to the
+%detected segment at iteration $i$ with width $w$.
+%In practice, the scan width is set a little greater than the assigned
+%width $\varepsilon$ ($k$ is a constant arbitrarily set to 4).
+where $C_{i-1}$, $\vec{D}_{i-1}$ and $w_{i-1}$ are a position, a director 
+vector and a width observed at iteration $i-1$.
+In the scope of the present detector, $C_{i-1}$ is the intersection of
+the input selection and the medial axis of $\mathcal{B}_{i-1}$,
+$\vec{D}_{i-1}$ the support vector of the narrowest digital straight line 
+that contains $\mathcal{B}_{i-1}$,
+and $w_{i-1}$ a value slightly greater than the minimal width of
+$\mathcal{B}_{i-1}$.
+So the last clause expresses the update of the scan bounds at iteration $i$.
 Compared to static directional scans, the scan strip moves while
 scan lines remain fixed.
 This behavior ensures a complete detection of the blurred segment even
@@ -145,15 +154,15 @@ when the orientation is badly estimated (\RefFig{fig:escape} c).
     \includegraphics[width=0.48\textwidth]{Fig_notions/escapeFirst_zoom.png} &
     \includegraphics[width=0.48\textwidth]{Fig_notions/escapeSecond_zoom.png} \\
     \multicolumn{2}{c}{
-    \includegraphics[width=0.98\textwidth]{Fig_notions/escapeThird_zoom.png}}
+    \includegraphics[width=0.78\textwidth]{Fig_notions/escapeThird_zoom.png}}
     \begin{picture}(1,1)(0,0)
       {\color{dwhite}{
-        \put(-260,134.5){\circle*{8}}
-        \put(-86,134.5){\circle*{8}}
+        \put(-260,108.5){\circle*{8}}
+        \put(-86,108.5){\circle*{8}}
         \put(-172,7.5){\circle*{8}}
       }}
-      \put(-263,132){a}
-      \put(-89,132){b}
+      \put(-263,106){a}
+      \put(-89,106){b}
       \put(-175,5){c}
     \end{picture}
   \end{tabular}
@@ -195,7 +204,7 @@ $\lambda$ iterations ($\mu_{i+\lambda} = \mu_i$), 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
+\varepsilon = \mu_{i+\lambda} + \frac{\textstyle 1}{\textstyle 2}
 \end{equation}
 This strategy aims at preventing the incorporation of spurious outliers in
 further parts of the segment.