Article: ours hunting and eq numbering

Article/expe.tex

@@ -12,8 +12,8 @@ For instance, the new directional scanners are encoded as an iterator so that
 only required scan lines are provided when required, whereas with the former
 code, all the scan lines were computed and returned, whenever they were used
 or not.
-We just checked that the outputs of the new coding of the former detector
-are equivalent to those of the old release.
+Outputs of the new coding of the former detector were compared to those of
+the old release, to ensure an equivalent behaviour.
 
 \begin{figure}[h]
 \center
@@ -42,7 +42,7 @@ are equivalent to those of the old release.
     \end{picture}
   \end{tabular}
   \caption{Outputs of both former (on left) and new (on right) detectors
-           using a selec tionof input strokes.}
+           using a selection of input strokes.}
   \label{fig:buro}
 \end{figure}
 
@@ -68,12 +68,13 @@ detectors with the input strokes of \RefFig{fig:buro}.}
 \label{tab:cmpOldNew}
 \end{table}
 
-In the second series of tests, we compare the execution times of both detectors
-for the automatic detection of edges on a set of test images. We display the
-results for one of them (\RefFig{fig:evalAuto}). X (resp. Y) blurred segments
-are extracted with the former (resp. new) detector on all images. The
-average execution time is X ms for the former detector, and Y ms for the
-new detector.
+In the second series of tests, the execution times of both detectors were
+compared on the automatic detection of edges on a set of test images.
+Results are displayed for one of them (\RefFig{fig:evalAuto}).
+X (resp. Y) blurred segments are extracted with the former
+(resp. new) detector on all images.
+The average execution time is X ms for the former detector,
+and Y ms for the new detector.
 
 \begin{figure}[h]
 \center

Article/method.tex

@@ -159,7 +159,7 @@ directions, so that in the exemple of figure (\RefFig{fig:edgeDir} b)),
 both edges (in different colours) are highlighted.
 These two thin blurred segments are much shorter, probably because the
 tiles are not perfectly aligned.
-This example illustrates our detector versatility.
+This example illustrates the versatility of the new detector.
 
 \subsection{Automatic blurred segment detection}
 

Article/notions.tex

@@ -4,9 +4,9 @@
 
 \subsection{Blurred segment}
 
-Our work relies on the notion of digital straight line as classically
+This work relies on the notion of digital straight line as classically
 defined in the digital geometry literature \cite{KletteRosenfeld04}.
-Only the 2-dimensional case is considered here.
+Only the 2D case is considered here.
 
 \begin{definition}
 A digital line $\mathcal{L}(a,b,c,\nu)$, with $(a,b,c,\nu) \in \mathbb{Z}^4$,
@@ -25,8 +25,8 @@ of points in $\mathbb{Z}^2$ that all belong to a digital line of
 arithmetical width $\varepsilon$.
 \end{definition}
 
-In this work, we use a linear-time algorithm that was developed to recognize
-a blurred segment of assigned width $\varepsilon$ \cite{DebledAl05}.
+A linear-time algorithm to recognize a blurred segment of assigned width
+$\varepsilon$ \cite{DebledAl05} is used in the work.
 It is based on an incremental growth of the convex hull of the blurred
 segment when adding each point successively.
 The minimal width $\mu$ of the blurred segment $\mathcal{B}$ is the
@@ -59,34 +59,36 @@ avoid the incorporation of spurious outliers in further parts of the segment.
 A directional scan is a partition of a digital straight line $\mathcal{D}$,
 called the {\it scan strip}, into scans $S_i$, each of them being a
 segment of a naive line $\mathcal{S}_i$ orthogonal to $\mathcal{D}$.
-\[ S_i = \mathcal{D} \cap \mathcal{S}_i,
-\mathcal{S}_i \perp \mathcal{D} \]
+\begin{equation}
+S_i = \mathcal{D} \cap \mathcal{S}_i, \mathcal{S}_i \perp \mathcal{D}
+\end{equation}
 The scans $S_i$ are developed on each side of a central scan $S_0$,
 and labelled with their manhattan distance ($d_1 = |d_x| + |d_y|$) to
 the central line $\mathcal{S}_0$ and a positive (resp.  negative) sign
-if their are on the left (resp. right) of $\mathcal{S}_0$.
+if they are on the left (resp. right) side of $\mathcal{S}_0$.
 
 \begin{figure}[h]
 \center
-  %\begin{picture}(300,40)
-  %\end{picture}
   \input{Fig_notions/fig}
   \caption{Example of directional scan.}
   \label{fig:ds}
 \end{figure}
 
-The directional scan can be defined by its central scan $S_0$.
+A directional scan can be defined by its central scan $S_0$.
 If $A(x_A,y_A)$ and $B(x_B,y_B)$ are the end points of $S_0$,
 the scan strip is defined by :
-\[ \mathcal{D}(A,B)
-= \mathcal{L}(\delta_x, \delta_y, min (c1,c2), 1 + |c_1-c_2|) \]
+\begin{equation}
+\mathcal{D}(A,B) = \mathcal{L}(\delta_x, \delta_y, min (c1,c2), 1 + |c_1-c_2|)
+\end{equation}
 \noindent
 where $\delta_x = x_B - x_A, \delta_y = y_B - y_A,
 c_1 = a\cdot x_A + b\cdot y_A$ and $c_2 = a\cdot x_B + b\cdot y_B$.
 
 The scan line $\mathcal{S}_i$ is then defined by :
-\[ \mathcal{S}_i(A,B) = \mathcal{L}(\delta_y, -\delta_x,
-\delta_y\cdot x_A - \delta_X\cdot y_A + i\cdot \nu_{AB}, \nu_{AB}) \]
+\begin{equation}
+\mathcal{S}_i(A,B) = \mathcal{L}(\delta_y, -\delta_x,
+\delta_y\cdot x_A - \delta_X\cdot y_A + i\cdot \nu_{AB}, \nu_{AB})
+\end{equation}
 where $\nu_{AB} = max (|\delta_x|, |\delta_y|)$
 
 The scan lines length is $d_\infty(AB)$ or $d_\infty(AB)-1$, where $d_\infty$
@@ -94,29 +96,32 @@ is the chessboard distance ($d_\infty = max (|d_x|,|d_y|)$).
 In practice, this difference of length between scan lines is not a drawback,
 as the image bounds should also be processed anyway.
 
-The directional scan can also be defined by its central point $C(x_C,y_C)$,
+A directional scan can also be defined by its central point $C(x_C,y_C)$,
 its direction $\vec{D}(x_D,y_D)$ and its width $w$. The scan strip is :
-\[ \mathcal{D}(C,\vec{D},w)
-= \mathcal{L}(y_D, -x_D, x_C\cdot y_D - y_C\cdot x_D - w / 2, w) \]
+\begin{equation}
+\mathcal{D}(C,\vec{D},w)
+= \mathcal{L}(y_D, -x_D, x_C\cdot y_D - y_C\cdot x_D - w / 2, w)
+\end{equation}
 
+\noindent
 and the scan line $\mathcal{S}_i(C,\vec{D},w)$ :
-\[ \mathcal{S}_i(C,\vec{D},w)
-= \mathcal{L}(x_D, y_D,
-              x_D\cdot x_C + y_D\cdot y_C - w / 2 + i\cdot w,
-              max (|x_D|,|y_D|) \]
+\begin{equation}
+\mathcal{S}_i(C,\vec{D},w) = \mathcal{L}(x_D, y_D,
+     x_D\cdot x_C + y_D\cdot y_C - w / 2 + i\cdot w, max (|x_D|,|y_D|)
+\end{equation}
 
 \subsection{Adaptive directional scan}
 
-We try to detect a blurred segment  inside a directional scan which position
+The blurred segment is searched inside a directional scan which position
 and orientation are given by the user, or defined in arbitrary direction in
 unsupervised mode.
 Most of the times, the detection stops where the segment escapes sideways
 froms the scan strip.
 Therefore a second search is run again using an other directional scan aligned
 on the detected segment.
-But we only have an estimation of this blurred segment direction, and the
-longer the real segment, the higher the probability to fail again on a
-blurred segment escape from the directional scan.
+But only an estimation of this blurred segment direction is provided,
+and the longer the real segment, the higher the probability to fail again
+on a blurred segment escape from the directional scan.
 
 \begin{figure}[h]
 \center
@@ -136,8 +141,8 @@ rather a conic shape to take into account the blurred segment preimage.
 % Of course this problem is amplified when ideal BS are considered.
 % -> FAUX : c'est relativement plus penalisant sur 1 BS reduit a 2 pts
 % que sur un BS qu'on n'autorise pas a s'elargir.
-This side shift is amplified when we let the blurred segment the capacity 
-to get thicker in order to capture possible noisy features.
+This side shift is amplified in situations where the blurred segment is
+left free to get thicker in order to capture possible noisy features.
 The assigned width is then still greater than the detected minimal width,
 so that the segment can move within the directional scan.
 Knowing the detected blurred segment shape and the image size, it is
@@ -172,10 +177,13 @@ but also of the multiple detection of the same segment start points.
   \label{fig:adaptiveScan}
 \end{figure}
 
-The solution we propose here is to dynamically adapt the scan direction
-on the detection result. At each position $i$, the scan strip is updated
-using the direction and minimal width of the blurred segment computed
-at the former position $i-1$, that is :
-\[ S_i = \mathcal{D} \cap \mathcal{S}_{i-1} \]
-Compared to static directional scans, the scan strip varies while the
+The proposed solution is to dynamically adapt the scan direction on the
+detection result.
+At each position $i$, the scan strip is updated using the direction and
+minimal width of the blurred segment computed at the former position $i-1$,
+that is :
+\begin{equation}
+S_i = \mathcal{D}_{i-1} \cap \mathcal{S}_i
+\end{equation}
+Compared to static directional scans, the scan strip is variable while the
 scan lines remain fixed.