diff --git a/Ipol/paper/Algos/algoFinal.tex b/Ipol/paper/Algos/algoFinal.tex
index f62c8773520078396d71b66cbac9c2a2ecc66215..31802f517abc286158f0c3bd2266a8cf24610cb6 100644
--- a/Ipol/paper/Algos/algoFinal.tex
+++ b/Ipol/paper/Algos/algoFinal.tex
@@ -7,7 +7,6 @@
\SetKwData{dsd}{$\vec{V}_0$}
\SetKwData{dirn}{$\vec{V}_G$}
\SetKwData{dire}{$\vec{V}_E$}
-\SetKwData{dst}{$W_0$}
\SetKwData{ath}{$\varepsilon$}
%\SetKwData{on}{isExtending}
%\SetKwData{nb}{$n_E$}
@@ -47,21 +46,22 @@
\SetKwFunction{rempt}{removePoint}
\SetKwFunction{thickness}{thickness}
\SetKwFunction{setath}{setMaxThickness}
-\SetKwFunction{ang}{angle}
+\SetKwFunction{ang}{$\angle$}
\SetKwFunction{dir}{direction}
\SetKwFunction{alignds}{alignScanner}
\SetKwFunction{centerline}{centerLine}
\Input{gradient magnitude map \gmap, gradient orientation map \vmap,
- directional scan center \dsc, direction \dsd and thickness \dst,
- initial assigned thickness \ath, edge gradient direction \dire}
+ directional scan center \dsc and direction \dsd,
+ maximal thickness \ath, reference edge gradient direction \dire,
+ maximal amount of successive detection fails \imax}
\Output{a detected blurred segment \bs}
\bs \takes $\emptyset$\;
\dirn \takes \ortho(\dsd, \dire)
\Comment*{Gets test gradient direction:
\dirn $\perp$ \dsd, \dirn$\cdot$\dire $> 0$}
-\ds \takes \getads (\dsc, \dsd, \dst) \Comment*{Gets an ADS from input scan.}
+\ds \takes \getads (\dsc, \dsd, 2\ath) \Comment*{Gets an ADS from input scan.}
\scan \takes \fscan (\ds)\;
\index \takes \locmax (\scan, \gmap, \gdiff, \vmap, \dirn)
\Comment*{Gets index of pixels with oriented gradient local max.}
@@ -75,16 +75,16 @@
\While{\ninter $-$ \ntemp $\neq$ (\imax, \imax)}{
\nb \takes \nb $+$ $1$\;
\bsth \takes \thickness(\bs)\;
- \If{\pinchcount = \pinchdelay}{
+ \If{\pinchcount $=$ \pinchdelay}{
\setath (\bs, \max (\ath, \thickness(\bs) + \half))\;
}
\If{\nb $>$ \adsdelay}{
- \If{\nb = \adsdelay $+$ $1$ and \angle(\dir(\bs),\dsd) $>$ \devang}{
+ \If{\nb $=$ \adsdelay $+$ $1$ and \ang(\dir(\bs),\dsd) $>$ \devang}{
\niside \takes \ntside $+$ (\imax, \imax)
\Comment*{Too large deviation}
}
\alignds(\ds,\centerline(\bs))
- \Comment*{Aligns scan stripe on blurred segment.}
+ \Comment*{Aligns scan strip on blurred segment.}
}
\ForEach{\side}{
\If{\niside $-$ \ntside $\leq$ \imax}{
diff --git a/Ipol/paper/Algos/algoInitial.tex b/Ipol/paper/Algos/algoInitial.tex
index 9bf3e43f1796781ecb51f31c214842840f810229..aa1cafefd08b20e78a94c10c271e1f6405d79ce4 100644
--- a/Ipol/paper/Algos/algoInitial.tex
+++ b/Ipol/paper/Algos/algoInitial.tex
@@ -28,8 +28,9 @@
\SetKwFunction{bsinit}{initializeBlurredSegment}
\SetKwFunction{addpt}{addPoint}
-\Input{gradient magnitude map \gmap,
- input segment \is, assigned thickness \ath}
+\Input{gradient magnitude map \gmap, input segment \is,
+ maximal thickness \ath,
+ maximal amount of successive detection fails \imax}
\Output{a detected blurred segment \bs}
\bs \takes $\emptyset$\;
diff --git a/Ipol/paper/Algos/algoSingle.tex b/Ipol/paper/Algos/algoSingle.tex
index ff7163bfb97bf2d008cdcdbcb5bebc4f9ff33f5a..85f378c666f8c3078675606cdf5403655504e254 100644
--- a/Ipol/paper/Algos/algoSingle.tex
+++ b/Ipol/paper/Algos/algoSingle.tex
@@ -9,7 +9,7 @@
\SetKwData{crossang}{$\alpha_{min}$}
\SetKwData{vecbs}{$\vec{D}$}
\SetKwData{nb}{$n$}
-\SetKwData{ang}{$angle$}
+\SetKwData{ang}{$\angle$}
\SetKwData{bsstart}{$P$}
\SetKwData{dire}{$\vec{V}_E$}
\SetKwData{centerpt}{$C$}
@@ -45,8 +45,8 @@
}{
\bsstart \takes \startpt (\bs)\;
\dire \takes \vmap $[$ \bsstart $]$\;
- \centerpt \takes \centerline (\bs) $\cup$ \is\;
- \bs \takes \finetrack (\gmap, \vmap, \centerpt, \vecbs, $2$ \ath, \ath, \dire)
+ \centerpt \takes \centerline (\bs) $\cap$ \is\;
+ \bs \takes \finetrack (\gmap, \vmap, \centerpt, \vecbs, \ath, \dire)
\Comment*{Runs fine tracking.}
\If{\size (\bs) $<$ \minsize}{
\bs \takes $\emptyset$ \Comment*{Not enough points.}
diff --git a/Ipol/paper/algos.tex b/Ipol/paper/algos.tex
index 8867825045b5405db4335eef67bb0b0ba1fc099a..b671b8dcca16f14f0e666090c1a8d9339b589674 100644
--- a/Ipol/paper/algos.tex
+++ b/Ipol/paper/algos.tex
@@ -28,15 +28,12 @@ Further extension is considered as useless.
On each side, the process stops after a fixed amount of successive extension
fails.
-\input{Algos/algoInitial}
-
-Three parameters are used in this step:
+Two parameters are used in this step:
a minimal gradient magnitude $G_{min}$ to select candidates,
-a maximal extent $N_E$ of the blurred segment,
-and a maximal count $N_I$ of successive extent fails.
-Nominal values are proposed for $G_{min}$ and $N_E$.
-$N_I$ is set to a default value,
-that can be adjusted according to the application context.
+a maximal extent $N_E$ of the blurred segment.
+Nominal values are proposed for them.
+
+\input{Algos/algoInitial}
\subsection{Fine tracking step of single extraction}
@@ -48,13 +45,11 @@ maxima of gradient magnitude found in the scan are tested untill a valid
one is found. Moreover the gradient orientation is checked when selecting
candidate points, using a rather strict maximal deviation value.
In order to avoid early stops when the detected segment escapes from the
-scan stripe, the directional scanner is aligned on the blurred segment
+scan strip, the directional scanner is aligned on the blurred segment
center line at each step. This alignment is only performed after a
predefined number of steps when the blurred segment direction gets
stable enough.
-\input{Algos/algoFinal}
-
Moreover when the growing segment stops thickenning, the assigned maximal
thickness of the recognition algorithm is pinched to a near value to
the observed thickness. This pinch procedure avoids further inclusion of
@@ -66,10 +61,12 @@ Finally, when any extension fail occurs, the amount of later successful
extensions must coincide to the fail count to validate these points.
This condition helps to detect more accurate segment ends.
-Three new parameters are used in this step:
+\input{Algos/algoFinal}
+
+Three parameters are used in this step:
a gradient height $\delta_G$ to discriminate local gradient maxima,
a minimal count $N_P$ of successive extents without blurred segment thickening,
-and a minimal extent $N_A$ before recentering the scan stripe.
+and a minimal extent $N_A$ before recentering the scan strip.
Nominal values are proposed for all of them.
\subsection{Single blurred segment extraction}
@@ -77,6 +74,9 @@ Nominal values are proposed for all of them.
The blurred segment extraction calls the initial detection, then the fine
tracking steps and applies optional validation tests to the output segment.
It is detailed in algorithm~/ref{algo:single}.
+Parameters are the maximal thickness $\varepsilon$ assigned to the blurred
+recognition algorithm, and the maximal amount $N_I$ of successive detection
+fail accepted.
\input{Algos/algoSingle}
diff --git a/Ipol/paper/article.tex b/Ipol/paper/article.tex
index f971cf004293aa151fb773089fd8d0e6ee830583..d004fcc368a3badeda461b3c21346a60671308e5 100644
--- a/Ipol/paper/article.tex
+++ b/Ipol/paper/article.tex
@@ -50,7 +50,7 @@
\SetCommentSty{mycmtsty}
% Use \newtheorem{} for remarks and definitions.
-\usepackage{amsthm}
+\usepackage{amsthm,amssymb}
\newtheorem{definition}{Definition}
\newtheorem*{remark}{Remark}
@@ -160,9 +160,9 @@ Compilation and execution instruction are included in the
%------------------------------------------------------------------------------
\section*{Acknowledgement}
-This work was originated in the scope of the French GeoDib ANR project
-("{\it G\'eom\'etrie des objets discrets bruit\'es}") held from 2006 to 2009
-and aiming at developing theoretical notions and algorithms to process noisy
+This work is a long-term outcome of French GeoDib ANR project
+("{\it G\'eom\'etrie des objets discrets bruit\'es}") held from 2006 to 2009,
+aiming at developing theoretical notions and algorithms to process noisy
discrete objects.
%------------------------------------------------------------------------------
diff --git a/Ipol/paper/figures/ads.pdf b/Ipol/paper/figures/ads.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..75cdc2b002d321bfdf6ebdc987c04946f87e4b4b
Binary files /dev/null and b/Ipol/paper/figures/ads.pdf differ
diff --git a/Ipol/paper/figures/bs.pdf b/Ipol/paper/figures/bs.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..445277891f872c5132a4bfc7bcc3d6a9cc772c5d
Binary files /dev/null and b/Ipol/paper/figures/bs.pdf differ
diff --git a/Ipol/paper/figures/ds.pdf b/Ipol/paper/figures/ds.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..bb25b61f799ab732ce2c489e283f7d9045d6ff10
Binary files /dev/null and b/Ipol/paper/figures/ds.pdf differ
diff --git a/Ipol/paper/figures/dsl.pdf b/Ipol/paper/figures/dsl.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..f808b05851bb98646a4b3c1e3ecca8d268530024
Binary files /dev/null and b/Ipol/paper/figures/dsl.pdf differ
diff --git a/Ipol/paper/notions.tex b/Ipol/paper/notions.tex
index 3afa12d7dc1feda6faca6837811b4987185eeff6..8e4a8c956fbae6d01b59f8b51dcd68c9d2f345bd 100644
--- a/Ipol/paper/notions.tex
+++ b/Ipol/paper/notions.tex
@@ -1,26 +1,172 @@
-\section{Required notions of discrete geometry}
+\section{Used notions of discrete geometry}
-Nouveau rappel des notions de geo discrete indispensables pour comprendre
-les algos :
-\begin{itemize}
-\item Discrete straight line
-\begin{itemize}
-\item{Thickness $\mu$}
-\end{itemize}
-\item Blurred segment
+In this section, the main notions of discrete geometry used in the
+framework are briefly recalled.
+
+A \textbf{digital straight line} $\mathcal{L}(a,b,c,\nu)$ is the set of
+points $p=(x,y) \in \mathbb{Z}^2$, such that:
+\begin{equation}
+0 \leq ax + by - c < \nu, \mbox{ with } (a,b,c,\nu) \in \mathbb{Z}^4
+\end{equation}
+The example of $\mathcal{L}(7,5,45,18)$ is given in \RefFig{fig:bs}(a).
+In this work, we use the following notations:
\begin{itemize}
-\item{Assigned thickness}
-\item{Optimal line}
-\item{Main lines of the recognition algorithm}
-\end{itemize}
-\item Directional scanner
-\item Adaptive directional scanner
+\item director vector: $\vec{V}(\mathcal{L}) = (a,b)$
+\item arithmetical width: $W(\mathcal{L}) = \nu$
+\item shift to origin: $H(\mathcal{L}) = c$
+\item period: $P(\mathcal{L}) = max(|a|,|b|)$
+\item lower bounding line:
+$L(\mathcal{L}) = \left\{(x',y')\in\mathbb{R}^2~|~ax'+by'=c\right\}$
+\item upper bounding line:
+$U(\mathcal{L}) = \left\{(x',y')\in\mathbb{R}^2~|~ax'+by'=c+\nu-1\right\}$
+\item thickness:
+$T(\mathcal{L})
+= \frac{\textstyle W(\mathcal{L})-1}{\textstyle P(\mathcal{L})}$.
\end{itemize}
+The thickness corresponds to the lowest value of the vertical and horizontal
+distances between lower and upper bounding lines.
+When $\nu = P(\mathcal{L})$, then $\mathcal{L}$ is the thinest 8-connected
+line and is called a \textit{naive line}.
+
+\begin{figure}[!htbp]
+\begin{center}
+ \begin{picture}(1,5)(0,0)
+% \put(0,0){\framebox(1,5)}
+ \end{picture}
+ \includegraphics[width=0.45\linewidth]{figures/dsl.pdf}
+ \begin{picture}(20,5)(0,0)
+% \put(0,0){\framebox(10,5)}
+ \put(-232,2){$O$}
+ \put(-12,2){$x$}
+ \put(-232,222){$y$}
+ \put(-115,-6){$a$}
+ \end{picture}
+ \includegraphics[width=0.45\textwidth]{figures/bs.pdf}
+ \begin{picture}(1,5)(0,0)
+% \put(0,0){\framebox(1,5)}
+ \put(-232,2){$O$}
+ \put(-12,2){$x$}
+ \put(-232,222){$y$}
+ \put(-115,-5){$b$}
+ \end{picture}
+ \caption{(a) Digital straight line $\mathcal{L}(7,5,45,18)$.
+ (b) Blurred segment of assigned thickness
+ $\varepsilon = \frac{17}{7}$, with convex hull in dashed
+ lines and optimal line $\mathcal{L}(7,5,45,18)$.}
+ \label{fig:bs}
+\end{center}
+\end{figure}
+
+A \textbf{digital straight segment} is a subset of a digital straight line
+restricted to $\left\{x~|~x\in [x_{min},x_{max}]\right\}$ if $|a|<|b|$,
+to $\left\{y~|~y\in [y_{min},y_{max}]\right\}$ otherwise.
+
+A \textbf{blurred segment} \cite{DebledAl06} $\mathcal{B}$ of assigned
+thickness $\varepsilon$ is a set of points in $\mathbb{Z}^2$ that belong
+to a digital straight line $\mathcal{L}$ of thickness
+$T(\mathcal{L}) \leq \varepsilon$ (see \RefFig{fig:bs}(b)).
+$\mathcal{L}$ is a \textit{covering line} of $\mathcal{B}$.
+The covering line with minimal thickness is called the
+\textit{optimal line} of $\mathcal{B}$.
+The thickness $T(\mathcal{B})$ of the blurred segment is the thickness
+of its optimal line.
+
+The blurred segment detection is based on a grawing procedure where input
+points are sequentially added and tested using a linear-time incremental
+recognition algorithm \cite{DebledAl06}. This algorithm relies on the
+maintenance of the blurred segment convex hull.
+
+A \textbf{directional scan} $\Delta_S$ is an ordered partition of the grid
+domain $\mathcal{G} \subset \mathbb{Z}^2$ restricted to a thick digital
+straight line $\mathcal{D}$, called \textit{scan strip},
+into naive segments $S_i$, called \textit{scans}, orthogonal to $\mathcal{D}$.
+Each scan $S_i$ is a segment of a naive line $\mathcal{N}_i$,
+called \textit{scan line}.
+The directional scan is defined as:
+\begin{equation}
+\Delta_S = \left\{ S_i = \mathcal{D} \cap \mathcal{N}_i \cap \mathcal{G}
+\left| \begin{array}{l}
+\vec{V}(\mathcal{N}_i) \cdot \vec{V}(\mathcal{D}) = 0 \\
+H(\mathcal{N}_i) = H(\mathcal{N}_{i-1}) + P(\mathcal{D})
+\end{array} \right. \right\}
+\end{equation}
+In this definition, $\vec{V}(\mathcal{N}_i) \cdot \vec{V}(\mathcal{D}) = 0$
+expresses the orthogonality between the scan lines $\mathcal{N}_i$
+and the scan strip $\mathcal{D}$.
+The shift $p(\mathcal{D})$ between successive scans $\mathcal{N}_{i-1}$
+and $\mathcal{N}_i$ guarantees that all points of $\mathcal{D}$ are traversed
+only one time.
+
+The directional scan is completely defined by grid domain $\mathcal{G}$
+and first scan $S_0$ (see \RefFig{fig:ads}(a)).
+In FBSD, $S_0$ can be defined either by its two end points $P_1$ and $P_2$,
+or by center point $C$, director vector $\vec{V}$ and length $L$.
+The scans $S_i$ can then be iteratively accessed from start scan $S_0$
+to last right and left scans.
+
+\begin{figure}[!htbp]
+\begin{center}
+ \begin{picture}(1,5)(0,0)
+% \put(0,0){\framebox(1,5)}
+ \end{picture}
+ \includegraphics[width=0.45\textwidth]{figures/ds.pdf}
+ \begin{picture}(20,5)(0,0)
+% \put(0,0){\framebox(10,5)}
+ \put(-236,-2){$O$}
+ \put(-12,-2){$x$}
+ \put(-236,234){$y$}
+ \put(-115,-6){$a$}
+ \put(-78,115){\vector(-4,-1){20}}
+ \put(-76,113){\makebox(4,4)[l]{$P_2$}}
+ \put(-208,46){\vector(4,1){20}}
+ \put(-214,44){\makebox(4,4)[r]{$P_1$}}
+ \end{picture}
+ \includegraphics[width=0.45\textwidth]{figures/ads.pdf}
+ \begin{picture}(1,5)(0,0)
+% \put(0,0){\framebox(1,5)}
+ \put(-236,-2){$O$}
+ \put(-12,-2){$x$}
+ \put(-236,234){$y$}
+ \put(-115,-5){$b$}
+ \put(-78,115){\vector(-4,-1){20}}
+ \put(-76,113){\makebox(4,4)[l]{$P_2$}}
+ \put(-208,46){\vector(4,1){20}}
+ \put(-214,44){\makebox(4,4)[r]{$P_1$}}
+ \put(-95,189){\vector(-1,0){40}}
+ \put(-93,186){\makebox(4,4)[l]{$\mathcal{C}$}}
+ \end{picture}
+ \caption{(a) A directional scan with start scan $S_0$ in black,
+ odd scans in green, even scans in blue,
+ bounds of scan lines $\mathcal{N}_i$ in dashed lines and
+ bounds of scan strip $\mathcal{D}$ in blue lines.
+ (b) An adaptive directional scan.
+ The scan strip is dynamically fit to curve $\mathcal{C}$ position,
+ and the scans are continuously centered on $\mathcal{C}$.}
+ \label{fig:ads}
+\end{center}
+\end{figure}
+
+An \textbf{adaptive directional scan} $\Delta_A$
+is a dynamical version of the directional scan with continuous registration
+of the scan strip to the center line $\mathcal{C}$ of a tracked linear feature.
+Compared to static directional scans, here the scan strip adapts from
+initial position $\mathcal{D}_0$ to update position $\mathcal{D}_i$,
+and at each step, the scan moves within its scan line
+(see \RefFig{fig:ads}(b)).
+The adaptive directional scan is formally defined by:
-Typically, The manuscript of an IPOL article submitted for peer-review should
-include an introduction, a detailed explanation of the algorithm (it could
-include some theoretical background, pseudo-code and/or block diagram, a
-complexity analysis, and an explanation of the parameter values), commented
-examples of the results and references to related publications. This content
-can be adapted when the article is not about an algorithm (like dataset
-articles).
+\begin{equation}
+\Delta_A = \left\{
+S_i = \mathcal{D}_i \cap \mathcal{N}_i \cap \mathcal{G}
+\left| \begin{array}{l}
+\vec{V}(\mathcal{N}_i) \cdot \vec{V}(\mathcal{D}_0) = 0 \\
+H(\mathcal{N}_i) = H(\mathcal{N}_{i-1}) + P(\mathcal{D}_0) \\
+\mathcal{D}_{i} = \mathcal{L}(\widehat{C}_i, W(\mathcal{D}_0)), i > 0
+\end{array} \right. \right\}
+\label{eq:ads}
+\end{equation}
+where $\widehat{C}_{i}$ is a triplet composed of the director vector
+$(a_i,b_i)$ and the shift to origin $c_i$ of an estimate at position $i$
+of the curve $\mathcal{C}$ to track.
+This last clause expresses the scan bounds update at iteration $i$.
+In FBSD, the tracked linear feature is the blurred segment to detect.