From 0aec1d29ccb4a60680aafa81c90335253e0c783b Mon Sep 17 00:00:00 2001
From: even <philippe.even@loria.fr>
Date: Tue, 4 Dec 2018 23:28:37 +0100
Subject: [PATCH] Article blurred segment width

---
 Article/Fig_notions/bswidth.tex | 21 +++++++++++++++++++++
 Article/biblio.bib              | 17 +++++++++++++++++
 Article/conclusion.tex          |  3 +--
 Article/intro.tex               | 17 ++++++++++++-----
 Article/main.tex                |  5 +++--
 Article/notions.tex             | 29 +++++++++++++++++------------
 6 files changed, 71 insertions(+), 21 deletions(-)
 create mode 100644 Article/Fig_notions/bswidth.tex

diff --git a/Article/Fig_notions/bswidth.tex b/Article/Fig_notions/bswidth.tex
new file mode 100644
index 0000000..4240fe9
--- /dev/null
+++ b/Article/Fig_notions/bswidth.tex
@@ -0,0 +1,21 @@
+\begin{picture}(220,60)
+  \multiput(0,6)(2,-6){2}{\line(3,1){150}}
+  \multiput(0,8)(30,6){8}{\line(5,1){10}}
+  \multiput(0,18)(30,6){8}{\line(5,1){10}}
+  \put(45,21){\circle*{3}}
+  \put(55,21){\circle*{3}}
+  \put(65,21){\circle*{3}}
+  \put(75,26){\circle*{3}}
+  \put(82.5,31){\circle*{3}}
+  \put(90,36){\circle*{3}}
+  \put(100,36){\circle*{3}}
+  \put(70,35){$\mathcal{B}$}
+  \put(110,30){\circle{3}}
+  \put(112,21){$P'$}
+  \put(140,36){\vector(0,1){10}}
+  \put(140,62.66){\vector(0,-1){10}}
+  \put(130,55){$\mu$}
+  \put(160,30){\vector(0,1){10}}
+  \put(160,60){\vector(0,-1){10}}
+  \put(164,44){$\mu '$}
+\end{picture}
diff --git a/Article/biblio.bib b/Article/biblio.bib
index 0f11aef..91c626c 100755
--- a/Article/biblio.bib
+++ b/Article/biblio.bib
@@ -85,3 +85,20 @@
   month="April",
   pages="11--12"
 }
+
+
+@InProceedings{NatsumiAl08,
+  author={Natsumi, Hiroaki and Sugimoto, Akihiro and Kenmochi, Yukiko},
+  title={Predicting corresponding region in a third view using
+         discrete epipolar lines.},
+  bookTitle={14th IAPR International Conference on Discrete Geometry
+             for Computer Imagery},
+  editor = {},
+  optaddress={Lyon, France},
+  pages={470--481},
+  month={April 16-18},
+  year={2008},
+  volume = {4992},
+  series = {LNCS},
+  publisher = {Springer}
+}
diff --git a/Article/conclusion.tex b/Article/conclusion.tex
index 125b6d9..484283f 100755
--- a/Article/conclusion.tex
+++ b/Article/conclusion.tex
@@ -21,5 +21,4 @@ Les filtres en fin de tracking sont l\`a pour soigner, pas pour gu\'erir.
 
 Perspectives : validation sur contextes applicatifs.
 
-\section*{Acknowledgements}
-This work was supported by La Cabane au Darou.
+%\section*{Acknowledgements}
diff --git a/Article/intro.tex b/Article/intro.tex
index f8aac74..6cec3ac 100755
--- a/Article/intro.tex
+++ b/Article/intro.tex
@@ -14,13 +14,20 @@ supervided context \cite{EvenMalavaud00}.
 Most of works aim at reducing their time complexity. }
 
 These methods rarely provide a direct measure of the quality of the output
-edge, such as sharpness, connectivity or scattering. Some information may
-often be drawn from their specific context, for example through
-an analysis of the peak in a Hough transform accumulator, or ...
+edge, such as sharpness, connectivity or scattering.
+Some information may often be drawn from their specific context, for example
+through an analysis of the peak in a Hough transform accumulator, or
+TO COMPLETE.
+In particular, the accuracy of the edge orientation may be quite critical
+in some application contexts, such as computer vision.
 
-In digital geometry, the notion of blurred segment \cite{Debled05,Buzer07}
+In digital geometry, the notion of blurred segment \cite{DebledAl05,Buzer07}
 was introduced to cope with the image noise or other sources of
-imperfections from the real world.
+imperfections from the real world. The pre-image of that geometrical object,
+ie the space of geometric entities which numerization matches this
+blurred segment, may convey useful information to evaluate possible moves in
+the 3D interpretations drawn, as a promising extension of former works
+on discrete epipolar geometry \cite{NatsumiAl08}.
 
 Our work aims at designing a flexible tool to detect such blurred segment
 in gray-level images for as well supervised as unsupervised contexts.
diff --git a/Article/main.tex b/Article/main.tex
index 3192451..6fa922e 100755
--- a/Article/main.tex
+++ b/Article/main.tex
@@ -15,8 +15,9 @@
 
 \begin{document}	
   \begin{frontmatter}
-    \title{Straight edge detection
-           based on an adaptive directional tracking of blurred segments}
+%    \title{Straight edge detection
+%           based on adaptive directional tracking of blurred segments}
+    \title{Adaptive directional tracking of blurred segments}
 
     \author{Philippe Even\inst{1} \and
             Phuc Ngo\inst{1} \and
diff --git a/Article/notions.tex b/Article/notions.tex
index 15023cb..5a2d709 100755
--- a/Article/notions.tex
+++ b/Article/notions.tex
@@ -16,29 +16,34 @@ When $\nu = max (|a|, |b|)$, $\mathcal{D}$ is the narrowest 8-connected
 line and is called a naive line.
 
 \begin{definition}
-A blurred segment of assigned width $\epsilon$ is a set $\mathcal{S}_\epsilon$
-of points in $\mathbb{Z}^2$ that all belong to a digital line of arithmetical
-width $\epsilon$.
+A blurred segment $\mathcal{B}$ of assigned width $\varepsilon$ is a set
+$\mathcal{S}_\varepsilon$ of points in $\mathbb{Z}^2$ that all belong
+to a digital line of arithmetical width $\varepsilon$.
 \end{definition}
 
 Linear time algorithms have been developed to recognize a blurred segment
-of assigned width $\epsilon$ \cite{DebledAl05,Buzer07}.
+of assigned width $\varepsilon$ \cite{DebledAl05,Buzer07}.
 They are 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 is the minimal width of
-this convex hull as computed by Melkman's algorithm \cite{Melkman87}.
-The extension of the blurred segment with a new input point is controlled
-by the recognition test $\mu < \epsilon$.
+The minimal width $\mu$ of the blurred segment $\mathcal{B}$ is the
+arithmetical width of the narrowest digital straight line that contains
+$\mathcal{B}$.
+It is also the minimal width of the convex hull, that is computed by
+Melkman's algorithm \cite{Melkman87}.
+The extension of the blurred segment with a new input point is thus
+controlled by the recognition test $\mu < \varepsilon$.
 
 \begin{figure}[h]
 \center
-  \begin{picture}(300,40)
-  \end{picture}
-  \caption{Example of blurred segment and recognition problem.}
+  \input{Fig_notions/bswidth}
+  \caption{A growing blurred segment $\mathcal{B}$ :
+when adding the new point $P'$, the blurred segment minimal width augments
+from $\mu$ to $\mu '$; if the new width $\mu '$ exceeds the assigned width
+$\varepsilon$, then the new input point is rejected.}
   \label{fig:bs}
 \end{figure}
 
-At the beginning, a large width $\epsilon_{ini}$ is assigned to the
+At the beginning, a large width $\varepsilon_{ini}$ is assigned to the
 recognition problem to allow the detection of large blurred segments.
 Then, when extending the blurred segment, this assigned width is
 gradually decremented to reach the detected blurred segment minimal width.
-- 
GitLab