From 4dc99b679c369e4fa1c732cf7aa7967669e8100a Mon Sep 17 00:00:00 2001
From: even <philippe.even@loria.fr>
Date: Wed, 12 Dec 2018 20:07:01 +0100
Subject: [PATCH] Article: blurred segments revisited
---
Article/intro.tex | 4 ++--
Article/notions.tex | 24 ++++++++++++++----------
2 files changed, 16 insertions(+), 12 deletions(-)
diff --git a/Article/intro.tex b/Article/intro.tex
index 6e472b7..f755bef 100755
--- a/Article/intro.tex
+++ b/Article/intro.tex
@@ -66,8 +66,8 @@ The search direction is fixed by the support vector of the blurred segment
detected at the former step, and because the set of vectors in a bounded
discrete space is finite, there is necessarily a limit on this direction
accuracy.
-It results that more steps would be necessary to process higher resolution
-images.
+It results that more steps would inevitably be necessary to process higher
+resolution images.
\subsection{Main contritions}
diff --git a/Article/notions.tex b/Article/notions.tex
index 3819553..a104d45 100755
--- a/Article/notions.tex
+++ b/Article/notions.tex
@@ -14,10 +14,11 @@ is the set of points $P(x,y)$ of $\mathbb{Z}^2$ that satisfy :
$0 \leq ax + by - c < \nu$.
\end{definition}
-$b/a$ is the slope of $\mathcal{L}$, $c$ its intercept and $\nu$
-its arithmetic width.
-When $\nu = max (|a|, |b|)$, $\mathcal{L}$ is the narrowest 8-connected
-line and is called a naive line.
+The parameters of the digital line are its slope $b/a$, its height $c$
+and its arithmetic width $\nu$. The set of points is organized as a
+periodic pattern of length $p = (|a|, |b|)$.
+When $\nu = p$, $\mathcal{L}$ is the narrowest 8-connected line
+and is called a naive line.
\begin{definition}
A blurred segment $\mathcal{B}$ of assigned width $\varepsilon$ is a set
@@ -32,8 +33,8 @@ segment when adding each point successively.
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}.
+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 $\mathcal{B}_i$ of assigned width
$\varepsilon$ and minimal width $\mu_i$ at step $i$ with a new input point
$P_{i+1}$ is thus controlled by the recognition test $\mu_{i+1} < \varepsilon$.
@@ -48,11 +49,14 @@ the assigned width $\varepsilon$, then the new input point is rejected.}
\label{fig:bs}
\end{figure}
-At the beginning, a large width $\varepsilon$ is assigned to the
+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, the assigned width is fixed to the observed minimal width in order to
-avoid the incorporation of spurious outliers in further parts of the segment.
+Then, when no more aumentation of the minimal width is observed as the
+segment grows ($\mu_{i+\lambda} = \mu_i$), the assigned width is fixed
+to the observed minimal width in order to avoid the incorporation of
+spurious outliers in further parts of the segment.
\subsection{Directional scan}
--
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