Article: figures referenced

Article/conclusion.tex

@@ -8,10 +8,10 @@ estimation of the edge thickness.
 It relies on directional scans of the image around maximal values of the
 gradient magnitude, that have previously been presented in
 \cite{KerautretEven09}.
-Despite of good performances obtained compared to existing detection methods
-found in the literature, the former approach suffers of two major drawbacks.
+Despite of good performances achieved, the former approach suffers of two
+major drawbacks.
 It does not estimate the edge thickness so that many outliers are inserted
-into the blurred segment and the provided estmation of the edge orientation 
+into the blurred segment and the provided estimation of the edge orientation 
 is biased.
 Then the scan direction is derived from a bounded blurred segment, that
 inevitably restricts its value to a finite set, so that long edges may be

Article/expe.tex

@@ -85,7 +85,8 @@ The former detector does not estimate the edge width, but just circumscribes
 the edge with a blurred segment of assigned width.
 If the edge is very thin, the blurred segment is free to rotate around the
 extracted edge and the provided orientation is biased.
-Moreover it lets some space to incorporate additional spurious outliers.
+Moreover it lets some space to incorporate additional spurious outliers,
+as illustrated in \RefFig{fig:outliers}.
 With the new appoach, a real estimation of the edge width is provided.
 The main risk of outlier incorporation remains at the beginning of the
 blurred segment expansion as long as the minimal width continues to grow

Article/method.tex

@@ -16,7 +16,7 @@ in the following figure.
     \put(24,0){\framebox(56,30)}
     \put(24,16){\makebox(56,10){Initial}}
     \put(24,4){\makebox(56,10){detection}}
-    \put(86,18){\scriptsize $\mathcal{B}_{1}$}
+    \put(86,18){\scriptsize $\mathcal{B}$}
     \put(80,15){\vector(1,0){22}}
     %\put(102,0){\framebox(56,30)}
     \multiput(102,15)(28,9){2}{\line(3,-1){28}}
@@ -29,10 +29,10 @@ in the following figure.
     \put(186,0){\framebox(56,30)}
     \put(186,16){\makebox(56,10){Fine}}
     \put(186,4){\makebox(60,10){tracking}}
-    \put(250,18){\scriptsize $\mathcal{B}_2$}
+    \put(250,18){\scriptsize $\mathcal{B}'$}
     \put(242,15){\vector(1,0){24}}
     \put(266,0){\framebox(56,30){Filtering}}
-    \put(330,18){\scriptsize $\mathcal{B}_3$}
+    \put(330,18){\scriptsize $\mathcal{B}''$}
     \put(322,15){\vector(1,0){22}}
   \end{picture}
   \caption{The detection process main workflow.}
@@ -40,7 +40,7 @@ in the following figure.
 \end{figure}
 
 The initial detection consists in building and extending a blurred segment
-$\mathcal{B}_1$ based on the highest gradient points found in each scan
+$\mathcal{B}$ based on the highest gradient points found in each scan
 of a static directional scanner based on an input segment $AB$.
 
 Validity tests aiming at rejecting too short or too sparse blurred segments
@@ -49,7 +49,7 @@ the position $C$ and direction $\vec{D}$ of this initial blurred segment
 are extracted.
 
 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
+$\mathcal{B}'$ 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.
 At this refinement step, the control of the assigned width is applied
@@ -58,7 +58,7 @@ direction $\vec{D}$ is used in order to extends the segment in the
 appropriate direction.
 
 The fine track output segment is finally filtered to remove artifacts
-and outliers, and a solution blurred segment $\mathcal{B}_3$ is provided.
+and outliers, and a final blurred segment $\mathcal{B}''$ is provided.
 
 \subsection{Adaptive directional scan}
 
@@ -134,7 +134,7 @@ 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} = 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
@@ -157,7 +157,7 @@ An example of adaptive directional scan is given in \RefFig{fig:adaption}.
            On the right picture, the scan bounds are displayed in red, the
            detected blurred segment in blue, and its bounding lines in green.
            The left picture displays the successive scans.
-           Adaption is quite sensible when crossing the tile joins.}
+           Here the adaption is visible at the crossing of the tile joins.}
   \label{fig:adaption}
 \end{figure}
 
@@ -204,7 +204,7 @@ i) the initial detection takes $M_j$ and the orthogonal direction $AB_\perp$
 to the stroke as input to build a static scan of fixed width
 $\varepsilon_{ini}$, and $M_j$ is used as start point of the blurred segment;
 ii) an occupancy mask, initially empty, is filled in with the points of the
-detected blurred segments $\mathcal{B}_j$ at the end of each successful
+detected blurred segments $\mathcal{B}_j''$ at the end of each successful
 detection;
 iii) points marked as occupied are rejected when selecting candidates for the
 blurred segment extension in the fine tracking step.
@@ -214,7 +214,8 @@ blurred segment extension in the fine tracking step.
 Beyond the possible detection of a large set of edges at once, the
 multi-detection allows the detection of some unaccessible edges in
 classical single detection mode. This is particularly the case of edges
-that are quite close to a more salient edge with a higher gradient.
+that are quite close to a more salient edge with a higher gradient,
+as illustrated in \RefFig{fig:voisins}.
 The multi-detection detects both edges and the user may then select
 the awaited one.
 

Article/notions.tex

@@ -38,7 +38,8 @@ arithmetical width of the narrowest digital straight line that contains
 $\mathcal{B}$.
 It is also the minimal width of the convex hull of $\mathcal{B}$,
 that can be computed by Melkman's algorithm \cite{Melkman87}.
-The extension of the blurred segment $\mathcal{B}_{i-1}$ of assigned width
+As depicted on \RefFig{fig:bs},
+the extension of the blurred segment $\mathcal{B}_{i-1}$ of assigned width
 $\varepsilon$ and minimal width $\mu_{i-1}$ at step $i-1$ with a new input
 point $P_i$ is thus controlled by the recognition test $\mu_i < \varepsilon$.
 
@@ -77,9 +78,9 @@ time.
 The scans $S_i$ are developed on each side of a start scan $S_0$,
 and ordered by their distance to the start line $\mathcal{N}_0$ with
 a positive (resp. negative) sign if they are on the left (resp. right)
-side of $\mathcal{N}_0$.
-The directional scan is iterately processedfrom the start scan to both ends.
-At each iteration $i$, the scans $S_i$ and $S_{-1}$ are successively processed.
+side of $\mathcal{N}_0$ (\RefFig{fig:ds}).
+The directional scan is iterately processed from the start scan to both ends.
+At each iteration $i$, the scans $S_i$ and $S_{-i}$ are successively processed.
 
 \begin{figure}[h]
 \center