Article: notions revisited

Article/Fig_notions/bswidth.tex

@@ -9,13 +9,13 @@
   \put(82.5,31){\circle*{3}}
   \put(90,36){\circle*{3}}
   \put(100,36){\circle*{3}}
-  \put(69,35){$\mathcal{B}_i$}
+  \put(66,37){$\mathcal{B}_{i-1}$}
   \put(110,30){\circle{3}}
-  \put(112,21){$P_{i+1}$}
+  \put(112,21){$P_i$}
   \put(140,36){\vector(0,1){10}}
   \put(140,62.66){\vector(0,-1){10}}
-  \put(129,56){$\mu_i$}
+  \put(120,58){$\mu_{i-1}$}
   \put(160,30){\vector(0,1){10}}
   \put(160,60){\vector(0,-1){10}}
-  \put(164,39){$\mu_{i+1}$}
+  \put(164,36){$\mu_i$}
 \end{picture}

Article/biblio.bib

@@ -131,7 +131,7 @@
 
 
 @inproceedings{LuAl15,
-  title = {CannyLines: A parameter-free line segment detector},
+  title = {CannyLines: a parameter-free line segment detector},
   author = {Lu, Xiaohu and Yao, Jian and Li, Kai and Li, Li},
   booktitle = {International Conference on Image Processing (ICIP)},
   publisher = {IEEE},

Article/intro.tex

@@ -20,8 +20,9 @@ to better fit to the discrete nature of most of todays data to process.
 In particular, 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 by the mean of a width parameter.
-Efficient algorithms were designed to recognize these digital objects in
-binary images.
+It is well suited to reflect the required edge quality information.
+Moreover efficient algorithms have already been designed to recognize
+these digital objects in binary images.
 
 The present work aims at designing a flexible tool to detect blurred segments
 with optimal width and orientation in gray-level images for as well
@@ -69,7 +70,7 @@ accuracy.
 It results that more steps would inevitably be necessary to process higher
 resolution images.
 
-\subsection{Main contritions}
+\subsection{Main contributions}
 
 The work presented in this paper aims at solving both former mentioned
 drawbacks through two main contributions:

Article/notions.tex

@@ -14,37 +14,40 @@ is the set of points $P(x,y)$ of $\mathbb{Z}^2$ that satisfy :
 $0 \leq ax + by - c < \nu$.
 \end{definition}
 
-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
+In the following, we note $\delta(\mathcal{L}) = b/a$ the slope of
+digital line $\mathcal{L}$, $w(\mathcal{L}) = \nu$ its arithmetical width,
+$h_P(\mathcal{L}) = c - ax - by$ its {\it shift} to point $P(x,y)$,
+$h_0(\mathcal{L}) = c$ its {\it shift} to origin, and
+$p(\mathcal{L}) = max(|a|,|b|)$ its period (i.e. the length of its
+periodic pattern).
+When $\nu = p$, then $\mathcal{L}$ 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
 of points in $\mathbb{Z}^2$ that all belong to a digital line of
-arithmetical width $\varepsilon$.
+arithmetical width $\nu = \varepsilon$.
 \end{definition}
 
 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.
+segment when adding each point $P_i$ 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 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 $\mu_{i+1} < \varepsilon$.
+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 $\mu_i < \varepsilon$.
 
 \begin{figure}[h]
 \center
   \input{Fig_notions/bswidth}
   \caption{A growing blurred segment $\mathcal{B}_i$ :
-when adding the new point $P_{i+1}$, the blurred segment minimal width
-augments from $\mu_i$ to $\mu_{i+1}$; if the new width $\mu_{+1}$ exceeds
+when adding the new point $P_i$, the blurred segment minimal width
+augments from $\mu_{i-1}$ to $\mu_i$; if the new width $\mu_i$ exceeds
 the assigned width $\varepsilon$, then the new input point is rejected.}
   \label{fig:bs}
 \end{figure}
@@ -54,22 +57,34 @@ 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 ($\mu_{i+\lambda} = \mu_i$), 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.
+segment grows ($\mu_{i+\lambda} = \mu_i$), the assigned width is set at
+a near value to the observed minimal width in order to avoid the
+incorporation of spurious outliers in further parts of the segment.
 
 \subsection{Directional scan}
 
-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 $\mathcal{D}$.
+A directional scan $DS$ is an ordered partition restricted to the image
+domain $\mathcal{I}$ 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 $\mathcal{N}_i$ orthogonal to $\mathcal{D}$.
 \begin{equation}
-S_i = \mathcal{D} \cap \mathcal{S}_i, \mathcal{S}_i \perp \mathcal{D}
+DS = \left\{ S_i = \mathcal{D} \cap \mathcal{N}_i \cap \mathcal{I} |
+\delta(\mathcal{N}_i) = - \delta^{-1}(\mathcal{D})
+\cap h_0(\mathcal{N}_i) = h_0(\mathcal{N}_{i-1}) + p(\mathcal{D}) \right\}
+%S_i = \mathcal{D} \cap \mathcal{N}_i, \mathcal{N}_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 they are on the left (resp. right) side of $\mathcal{S}_0$.
+In this expression, the clause
+$\delta(\mathcal{N}_i) = - \delta^{-1}(\mathcal{D})$
+expresses the othogonality constraint between the scan lines $\mathcal{N}_i$
+and the scan strip $\mathcal{D}$.
+Then the shift of the period $p(\mathcal{D})$ between successive scans
+guarantees that all points of the scan strip are travelled one and only one
+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 $\mathcal{N}_0$.
 
 \begin{figure}[h]
 \center
@@ -78,7 +93,7 @@ if they are on the left (resp. right) side of $\mathcal{S}_0$.
   \label{fig:ds}
 \end{figure}
 
-A directional scan can be defined by its central scan $S_0$.
+A directional scan can be defined by its start 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 :
 \begin{equation}
@@ -88,17 +103,17 @@ the scan strip is defined by :
 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 :
+The scan line $\mathcal{N}_i$ is then defined by :
 \begin{equation}
-\mathcal{S}_i(A,B) = \mathcal{L}(\delta_y, -\delta_x,
+\mathcal{N}_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 $d_\infty$
-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 scan lines length is $d_\infty(AB)$ or $d_\infty(AB)-1$, where $d_\infty$
+%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.
 
 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 :
@@ -108,86 +123,98 @@ its direction $\vec{D}(x_D,y_D)$ and its width $w$. The scan strip is :
 \end{equation}
 
 \noindent
-and the scan line $\mathcal{S}_i(C,\vec{D},w)$ :
+and the scan line $\mathcal{N}_i(C,\vec{D},w)$ :
 \begin{equation}
-\mathcal{S}_i(C,\vec{D},w) = \mathcal{L}(x_D, y_D,
+\mathcal{N}_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}
 
-The blurred segment is searched inside a directional scan which position
+The blurred segment is searched within 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
+froms the scan strip (\RefFig{fig:escape}).
+Therefore a second search is run again using a second directional scan aligned
 on the detected segment.
-But only an estimation of this blurred segment direction is provided,
+But only a quantized estimation of this blurred segment direction is given,
 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
-  \begin{picture}(300,40)
-  \end{picture}
+  \begin{tabular}{c}
+    \includegraphics[width=0.49\textwidth]{Fig_notions/escapeFirst_zoom.png}
+  \end{tabular}
   \caption{Example of early detection failures
            on side escapes from the directional scan.}
-  \label{fig:sideEscapes}
+  \label{fig:escape}
 \end{figure}
 
-Even in ideal situation where the detected segment is a perfect line,
-its width is never null as a result of the discretization process.
-The estimated direction accuracy is mostly constrained by the length of
-the detected segment.
-To avoid these side escapes, the scan should not be a linear strip but
-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 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
-possible to define a conic scan area, but this solution is computationaly
-expensive because it leads to useless exploration of large image areas.
-
-\begin{figure}[h]
-\center
-  %\begin{picture}(300,40)
-  %\end{picture}
-  \input{Fig_notions/bscone}
-  \caption{Possible extension area based
-           on the detected blurred segment preimage.}
-  \label{fig:cone}
-\end{figure}
-
-The solution implemented in the former work was to let some arbitrary
-margin between the scan strip width and the assigned width to the detection,
-and to perform two fine detection steps, using for each of them the direction
-found at the former step.
+%Even in ideal situation where the detected segment is a perfect line,
+%its width is never null as a result of the discretization process.
+%The estimated direction accuracy is mostly constrained by the length of
+%the detected segment.
+%To avoid these side escapes, the scan should not be a linear strip but
+%rather a conic shape to take into account the blurred segment preimage.
+%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
+%possible to define a conic scan area, but this solution is computationaly
+%expensive because it leads to useless exploration of large image areas.
+%
+%\begin{figure}[h]
+%\center
+%  %\begin{picture}(300,40)
+%  %\end{picture}
+%  \input{Fig_notions/bscone}
+%  \caption{Possible extension area based
+%           on the detected blurred segment preimage.}
+%  \label{fig:cone}
+%\end{figure}
+
+In the former work, an additional refinement step was run,
+but doing so, the problem was just delayed further.
+%The solution implemented in the former work was to let some arbitrary
+%margin between the scan strip width and the assigned width to the detection,
+%and to perform two fine detection steps, using for each of them the direction
+%found at the former step.
 This process could be itered as long as the blurred segment escapes from
-the directional scanner using as any detection steps as necessary.
-But it produces a useless computational coast, because of the margin left,
-but also of the multiple detection of the same segment start points.
+the directional scanner using as any fine detection steps as necessary.
+But the multiple detection of the same segment start points produces
+a useless computational coast.
+
+Here the proposed solution is to dynamically adapt the scan direction on
+the detection result.
+At each position $i+1$, the scan strip is updated using the direction
+of the blurred segment computed at the former position $i$.
+The adaptive directional scan $ADS$ is then defined by :
+\begin{equation}
+%S_i = \mathcal{D}_{i-1} \cap \mathcal{N}_i
+ADS = \left\{ S_i = \mathcal{D}_i \cap \mathcal{N}_i \cap \mathcal{I} |
+\delta(\mathcal{N}_i) = - \delta^{-1}(\mathcal{D}_0)
+\cap h_0(\mathcal{N}_i) = h_0(\mathcal{N}_{i-1}) + p(\mathcal{D})
+\cap \mathcal{D}_{i+1} = d (\mathcal{B}_i,\varepsilon) \right\}
+\end{equation}
+The last clause expresses the update of the scan bounds at step $i+1$.
+Compared to static directional scans, the scan strip moves while
+scan lines remain fixed.
+An example of adaptive directional scan is given in \RefFig{fig:adaption}.
 
 \begin{figure}[h]
 \center
-  \begin{picture}(300,40)
-  \end{picture}
+  \begin{tabular}{c@{\hspace{0.2cm}}c}
+    \includegraphics[width=0.49\textwidth]{Fig_notions/adaptionBounds_zoom.png} &
+    \includegraphics[width=0.49\textwidth]{Fig_notions/adaptionLines_zoom.png}
+  \end{tabular}
   \caption{Example of blurred segment detection
-           using an adaptive directional scan.}
-  \label{fig:adaptiveScan}
+           using an adaptive directional scan.
+           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.}
+  \label{fig:adaption}
 \end{figure}
-
-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.