Article: first global review

Article/Fig_method/workflow.tex

@@ -10,7 +10,7 @@
     %\put(102,0){\framebox(56,30)}
     \multiput(102,15)(28,9){2}{\line(3,-1){28}}
     \multiput(102,15)(28,-9){2}{\line(3,1){28}}
-    \put(100,0){\makebox(60,30){Valid ?}}
+    \put(100,0){\makebox(60,28){Valid ?}}
     \put(133,-2){\scriptsize $\emptyset$}
     \put(130,6){\vector(0,-1){10}}
     \put(159,18){\scriptsize $(C,\vec{D})$}
@@ -20,7 +20,12 @@
     \put(186,4){\makebox(60,10){tracking}}
     \put(250,18){\scriptsize $\mathcal{B}'$}
     \put(242,15){\vector(1,0){24}}
-    \put(266,0){\framebox(56,30){Filtering}}
-    \put(330,18){\scriptsize $\mathcal{B}''$}
+%    \put(266,0){\framebox(56,30){Filtering}}
+    \multiput(266,15)(28,9){2}{\line(3,-1){28}}
+    \multiput(266,15)(28,-9){2}{\line(3,1){28}}
+    \put(264,0){\makebox(60,28){Accept ?}}
+    \put(297,-2){\scriptsize $\emptyset$}
+    \put(294,6){\vector(0,-1){10}}
+    \put(330,18){\scriptsize $\mathcal{B}'$}
     \put(322,15){\vector(1,0){22}}
   \end{picture}

Article/expe.tex

@@ -2,17 +2,87 @@
 
 \label{sec:expe}
 
-The evaluation stage aims at quantifying the advantages of the new detector
-compared to the former one.
+The main goal of this work is to provide straight segments with a quality
+indication through the associated width parameter.
+In lack of available reference tool, the evaluation stage mostly aims
+at quantifying the advantages of the new detector compared to the previous
+detector in unsupervised context.
 For a fair comparison, the process flow of the former method (the initial
 detection followed by two refinement steps) is integrated as an option
 into the code of the new detector, so that both methods rely on the same
 optimized basic routines.
 
-\input{expeSynthese}
+The first test (\RefFig{fig:synth}) compares the performance of both
+detectors on a set of 1000 synthesized images containing 10 randomly
+placed input segments with random width between 1 and 4 pixels.
+Altough this perfect world context with low gradient noise tends to soften
+the old detector weaknesses, the results of \RefTab{tab:synth} show slightly
+better width and angle measurements on long segments for the new detector.
+The new detector generates more small segments that degrade the angle
+estimations, but it produces a smaller amount of false detections and
+succeeds in finding most of the input segments.
 
-\input{expeHard}
+The second test (\RefFig{fig:hard}) visually compares the results of
+both detectors on quite difficult images, even for other detectors from
+the literature, with a lot of gradient noise. 
+The new detector provides less outliers and misaligned segments, and
+globally more relevant informations to infere the structure of the brick wall.
+\begin{figure}[h]
+%\center
+  \begin{tabular}{
+      c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}c}
+    \includegraphics[width=0.19\textwidth]{Fig_synth/statsExample.png} &
+    \includegraphics[width=0.19\textwidth]{Fig_synth/statsoldPoints.png} &
+    \includegraphics[width=0.19\textwidth]{Fig_synth/statsoldBounds.png} &
+    \includegraphics[width=0.19\textwidth]{Fig_synth/statsnewPoints.png} &
+    \includegraphics[width=0.19\textwidth]{Fig_synth/statsnewBounds.png}
+    \begin{picture}(1,1)
+      \put(-310,0){a)}
+      \put(-240,0){b)}
+      \put(-170,0){c)}
+      \put(-100,0){d)}
+      \put(-30,0){e)}
+    \end{picture}
+  \end{tabular}
+  \caption{Evaluation on synthesized images:
+           a) one of the test images,
+           b) output blurred segments from the old detector and
+           c) their enclosing digital segments,
+           d) output blurred segments from the new detector and
+           e) their enclosing digital segments.}
+  \label{fig:synth}
+\end{figure}
+\begin{table}
+\centering
+\input{Fig_synth/statsTable}
+\caption{Measured performance of both detectors on a set of synthesized images.
+$S$ is the set of all the input segments,
+$D$ the set of all the detected blurred segments.}
+\label{tab:synth}
+\end{table}
+\begin{figure}
+%\center
+  \begin{tabular}{
+      c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}}
+    \includegraphics[width=0.32\textwidth]{Fig_method/parpaings.png} &
+    \includegraphics[width=0.32\textwidth]{Fig_hard/hardOld.png} &
+    \includegraphics[width=0.32\textwidth]{Fig_hard/hardNew.png}
+    \begin{picture}(1,1)
+      {\color{dwhite}{
+        \put(-286,4.5){\circle*{8}}
+        \put(-171,4.5){\circle*{8}}
+        \put(-58,4.5){\circle*{8}}
+      }}
+      \put(-288.5,2){a}
+      \put(-173.5,2){b}
+      \put(-60.5,2){c}
+    \end{picture}
+  \end{tabular}
+  \caption{Detection results on quite textured images:
+           one of the tested images (a),
+           the segments found by the old detector (b)
+           and those found by the new detector (c).}
+  \label{fig:hard}
+\end{figure}
 
 \input{expeAuto}
-
-%\input{expeOld}

Article/expeAuto.tex

 
-The third series of tests aims at evaluating the performance of the new
+The third series of tests aim at evaluating the performance of the new
 detector wrt the previous one on a selection of more standard images.
 Compared measures $M$ are the execution time $T$, the amount $N$ of detected
 blurred segments, the amount $N'$ of long (larger than 40 pixels) segments,
 the mean length $L$ and the mean width $W$ of the detected segments.
-In order to be objective, these results are also compared to the same
+For the sake of objectivity, these results are also compared to the same
 measurements made on the image data base used for the CannyLine line
 segment detector \cite{LuAl15}.
 The table \RefTab{tab:auto} gives the measures obtained on one of the
 selected images (\RefFig{fig:auto}) and the result of a systematic test
 on the whole CannyLine data base.
-The new detector is faster and provides more segments.
-These segments are mainly shorter and thinner.
-The control of the assigned width to fit to the detected segment width
-has the side effect of blocking the segment expansion when the remote parts
-are more noisy.
-The relevance of this behavior depends strongly on application requirements.
 \begin{figure}[h]
 %\center
   \begin{tabular}{
       c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}}
     \includegraphics[width=0.32\textwidth]{Fig_auto/buro.png} &
     \includegraphics[width=0.32\textwidth]{Fig_auto/autoOld.png} &
-    \includegraphics[width=0.32\textwidth]{Fig_auto/autoNew.png}
+    \includegraphics[width=0.32\textwidth]{Fig_auto/autoNew.png} \\
+    & \includegraphics[width=0.22\textwidth]{Fig_auto/dssDetailOld.png} &
+    \includegraphics[width=0.22\textwidth]{Fig_auto/dssDetailNew.png}
     \begin{picture}(1,1)
+      {\color{red}{
+        \put(-21.5,38){\framebox(18,8)}
+        \put(-12.5,38){\vector(-2,-1){25}}
+        \put(-135.5,38){\framebox(18,8)}
+        \put(-124.5,38){\vector(-2,-1){25}}
+      }}
       {\color{dwhite}{
-        \put(-286,4.5){\circle*{8}}
-        \put(-171,4.5){\circle*{8}}
-        \put(-58,4.5){\circle*{8}}
+        \put(-287,37.5){\circle*{8}}
+        \put(-172,37.5){\circle*{8}}
+        \put(-59,37.5){\circle*{8}}
+        \put(-186,5.5){\circle*{8}}
+        \put(-73,5.5){\circle*{8}}
       }}
-      \put(-288.5,2){a}
-      \put(-173.5,2){b}
-      \put(-60.5,2){c}
+      \put(-289.5,35){a}
+      \put(-174.5,35){b}
+      \put(-61.5,35){c}
+      \put(-189,3){d}
+      \put(-75.5,3){e}
     \end{picture}
   \end{tabular}
-  \caption{Automatic detection of segments:
-           a) input image,
-           b) results of the old detector,
-           c) results of the new detector.}
+  \caption{Automatic detection on standard images:
+           an input image (a), the segments found by the old detector (b)
+           and those found by the new detector (c), and a detail of the
+           enclosing digital segments for both old (d) and new (e) detectors.}
   \label{fig:auto}
 \end{figure}
 \begin{table}
@@ -48,3 +54,22 @@ The relevance of this behavior depends strongly on application requirements.
          the previous (resp. new) detector.}
 \label{tab:auto}
 \end{table}
+
+The new detector is faster and provides more blurred segments than the
+previous one.
+The details of \RefFig{fig:auto} d) and e) illustrate the improved
+accuracy obtained.
+The output segments are thinner but also shorter.
+The control of the assigned width to fit to the detected segment width
+has the side effect of blocking the segment expansion when the remote parts
+are more noisy.
+The relevance of this behavior depends strongly on application requirements.
+Therefore the control of the assigned width is left as an option the user
+can let or cancel.
+In both case, it could be interesting to combine the detector with a tool
+to merge aligned segments.
+
+%Although these observations in unsupervised context should be reproduced
+%in supervised context, similar experiments require an application context
+%to dispose of a ground truth and of real users to assess the detector
+%relevance through ergonomy evaluations.

Article/expeHard.tex

@@ -22,8 +22,8 @@ globally more relevant informations to infere the structure of the brick wall.
     \end{picture}
   \end{tabular}
   \caption{Detection results on quite textured images:
-           1) one of the tested images,
-           2) the segments found by the old detector,
-           3) and the one found by the new detector.}
+           one of the tested images (a),
+           the segments found by the old detector (b)
+           and those found by the new detector (c).}
   \label{fig:hard}
 \end{figure}

Article/expeSynthese.tex

@@ -27,17 +27,17 @@ succeeds in finding most of the input segments.
   \end{tabular}
   \caption{Evaluation on synthesized images:
            a) one of the test images,
-           b) output segments points from the old detector and
-           c) their minimal digital straight segments,
-           d) output segments points from the new detector and
-           e) their minimal digital straight segments.}
+           b) output blurred segments from the old detector and
+           c) their enclosing digital segments,
+           d) output blurred segments from the new detector and
+           e) their enclosing digital segments.}
   \label{fig:synth}
 \end{figure}
 \begin{table}
 \centering
 \input{Fig_synth/statsTable}
 \caption{Measured performance of both detectors on a set of synthesized images.
-$S$ is the set of all the input segments pixels,
-$D$ the set of all the detected segments pixels.}
+$S$ is the set of all the input segments,
+$D$ the set of all the detected blurred segments.}
 \label{tab:synth}
 \end{table}

Article/intro.tex

@@ -34,11 +34,9 @@ or at least quite few values with intuitive meaning.
 
 In a former paper \cite{KerautretEven09}, an efficient tool to detect
 blurred segments of fixed width in gray-level images was already introduced.
-It is based on a first rough detection in a local area
-of the image either defined by the user in supervised context or blindly
-explored in automatic mode. The goal is to disclose the presence of an edge.
-Therefore a simple test as the gradient maximal value is performed.
-
+It is based on a first rough detection in a local image area
+defined by the user. The goal is to disclose the presence of a straight edge.
+Therefore an as simple test as the gradient maximal value is performed.
 In case of success, refinement steps are run through an exploration of
 the image in the direction of the detected edge.
 In order to prevent local disturbances such as the presence of a sharper
@@ -57,22 +55,20 @@ user according to the application requirements. The produced information
 on the edge quality is rather poor, and especially when the edge is thin,
 the risk to incorporate outlier points is quite high, thus producing a
 biased estimation of the edge orientation.
-
 Then, two refinement steps are systematically run.
 On one hand, this is useless when the first detection is successfull.
 On the other hand, there is no guarantee that this approach is able to
 process larger images.
 The search direction relies on the support vector of the blurred segment
-detected at the former step, and the numerization rounding fixes a limit
-on this estimated orientation accuracy.
-It results that more steps would inevitably be necessary to process higher
-resolution images.
+detected at the former step.
+Because the numerization rounding fixes a limit on this estimated orientation
+accuracy, more steps are inevitably necessary to process larger images.
 
 \subsection{Main contributions}
 
 The work presented in this paper aims at solving both former mentioned
 drawbacks through two main contributions:
-(i) the concept of adaptive directional scanner designed to get some
+(i) the concept of adaptive directional scan designed to get some
 compliance to the unpredictable orientation problem;
 (ii) the control of the assigned width to the blurred segment recognition
 algorithm, intended to derive more reliable information on the edge
@@ -84,9 +80,9 @@ which can be evaluated through an online demonstration.
 
 In the next section, the main theoretical notions this work relies on are
 introduced.
-The new detector workflow, the adaptive directional scanner, the control
+The new detector workflow, the adaptive directional scan, the control
 of the assigned with and their integration into both supervised and
-unsupervised contexts are then presented and discussed in \RefSec{sec:method}.
+unsupervised contexts are then presented in \RefSec{sec:method}.
 Experiments led to assess the expected increase of performance are decribed
 in \RefSec{sec:expe}.
 Finally, achieved results are summarized in \RefSec{sec:conclusion},

Article/method.tex

@@ -4,8 +4,7 @@
 
 \subsection{Workflow of the detection process}
 
-The workflow of the blurred segment detection process is summerized
-in the following figure.
+The workflow of the detection process is summerized in the following figure.
 
 \begin{figure}[h]
 \center
@@ -16,11 +15,11 @@ in the following figure.
 
 The initial detection consists in building and extending a blurred segment
 $\mathcal{B}$ based on the highest gradient points found in each scan
-of a static directional scanner based on an input segment $AB$.
+of a static directional scan based on an input segment $AB$.
 
 Validity tests are then applied to decide of the detection poursuit.
 They aim at rejecting a too short or too sparse blurred segment, or a
-blurred segment with a close orientation to the input segment $AB$.
+blurred segment with a too close orientation to the input segment $AB$.
 In case of positive response, the position $C$ and direction $\vec{D}$
 of this initial blurred segment are extracted.
 
@@ -29,13 +28,17 @@ $\mathcal{B}'$ based on points that correspond to local maxima of the
 image gradient, ranked by magnitude order, and with gradient direction
 close to a reference gradient direction at the segment first point.
 At this refinement step, a control of the assigned width is applied
-and an adaptive directional scanner based on the found position $C$ and
+and an adaptive directional scan based on the found position $C$ and
 direction $\vec{D}$ is used in order to extends the segment in the
 appropriate direction. These two improvements are described in the
 following sections.
 
-The fine track output segment is finally filtered to remove artifacts
-and outliers, and a final blurred segment $\mathcal{B}''$ is provided.
+The output segment $\mathcal{B}'$ is finally tested according to the
+application needs. Too short, too sparse ot too fragmented segments
+can be rejected. Length, sparsity or fragmentation thresholds are
+intuitive parameters left at the end user disposal.
+None of these tests are activated for the experimental stage in order
+to put forward achievable perfomance.
 
 \subsection{Adaptive directional scan}
 
@@ -53,125 +56,75 @@ is subject to the numerization rounding, and the longer the real segment
 to detect, the higher the probability to fail again on a blurred segment
 escape from the directional scan.
 
-%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}
+\begin{figure}[h]
+\center
+  \begin{tabular}{c@{\hspace{0.2cm}}c}
+    \includegraphics[width=0.48\textwidth]{Fig_notions/escapeFirst_zoom.png} &
+    \includegraphics[width=0.48\textwidth]{Fig_notions/escapeSecond_zoom.png} \\
+    \multicolumn{2}{c}{
+    \includegraphics[width=0.62\textwidth]{Fig_notions/escapeThird_zoom.png}}
+    \begin{picture}(1,1)(0,0)
+      {\color{dwhite}{
+        \put(-260,87.5){\circle*{8}}
+        \put(-86,87.5){\circle*{8}}
+        \put(-172,7.5){\circle*{8}}
+      }}
+      \put(-263,85){a}
+      \put(-89,85){b}
+      \put(-175,5){c}
+    \end{picture}
+  \end{tabular}
+  \caption{Aborted detections on side escapes of static directional scans
+           and successful detection using an adaptive directional scan.
+           The last points added to the left of the blurred segment during
+           initial detection (a) lead to a bad estimation of its
+           orientation, and thus to an incomplete fine detection with a
+           classical directional scan (b). An adaptive directional scan
+           can continue the segment expansion as far as necessary (c).
+           On the pictures, the input selection is drawn in red color,
+           the scan strip bounds
+           in blue and the detected blurred segment in green.}
+  \label{fig:escape}
+\end{figure}
 
 To overcome this issue, in the former work, an additional refinement step is
 run using the better orientation estimated from the longer segment obtained.
 It is enough to completely detect most of the tested edges, but certainly
 not all, especially if larger images with much longer edges are processed.
-%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.
 As a solution, this operation could be itered as long as the blurred segment
-escapes from the directional scanner using as any fine detection steps as
+escapes from the directional scan using as any fine detection steps as
 necessary.
 But at each iteration, already tested points are processed again,
 thus producing a useless computational cost.
 
 Here the proposed solution is to dynamically align the scan direction on
-the blurred segment one all along the expansion stage.
+the blurred segment all along the expansion stage.
 At each iteration $i$ of the expansion, the scan strip is aligned on the
 direction of the blurred segment $\mathcal{B}_{i-1}$ computed at previous
 iteration $i-1$.
 More generally, an adaptive directional scan $ADS$ is 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}
 \left| \begin{array}{l}
 \delta(\mathcal{N}_i) = - \delta^{-1}(\mathcal{D}_0) \\
-\wedge~ h_0(\mathcal{N}_i) = h_0(\mathcal{N}_{i-1}) + p(\mathcal{D}) \\
+\wedge~ h(\mathcal{N}_i) = h(\mathcal{N}_{i-1}) + p(\mathcal{D}) \\
 \wedge~ \mathcal{D}_{i} = \mathcal{D} (C_{i-1}, \vec{D}_{i-1}, w_{i-1}), i > 1
-%\wedge~ \mathcal{D}_{i} = D (\mathcal{B}_{i-1},\varepsilon + k), i > 1
 \end{array} \right. \right\}
 \end{equation}
-%where $D (\mathcal{B}_i,w)$ is the scan strip aligned to the
-%detected segment at iteration $i$ with width $w$.
-%In practice, the scan width is set a little greater than the assigned
-%width $\varepsilon$ ($k$ is a constant arbitrarily set to 4).
 where $C_{i-1}$, $\vec{D}_{i-1}$ and $w_{i-1}$ are a position, a director 
 vector and a width observed at iteration $i-1$.
 In the scope of the present detector, $C_{i-1}$ is the intersection of
 the input selection and the medial axis of $\mathcal{B}_{i-1}$,
-$\vec{D}_{i-1}$ the support vector of the narrowest digital straight line 
-that contains $\mathcal{B}_{i-1}$,
-and $w_{i-1}$ a value slightly greater than the minimal width of
-$\mathcal{B}_{i-1}$.
+$\vec{D}_{i-1}$ the support vector of the enclosing digital segment 
+$E(\mathcal{B}_{i-1})$, and $w_{i-1}$ a value slightly greater than the
+minimal width of $\mathcal{B}_{i-1}$.
 So the last clause expresses the update of the scan bounds at iteration $i$.
 Compared to static directional scans, the scan strip moves while
 scan lines remain fixed.
 This behavior ensures a complete detection of the blurred segment even
 when the orientation is badly estimated (\RefFig{fig:escape} c).
 
-\begin{figure}[h]
-\center
-  \begin{tabular}{c@{\hspace{0.2cm}}c}
-    \includegraphics[width=0.48\textwidth]{Fig_notions/escapeFirst_zoom.png} &
-    \includegraphics[width=0.48\textwidth]{Fig_notions/escapeSecond_zoom.png} \\
-    \multicolumn{2}{c}{
-    \includegraphics[width=0.72\textwidth]{Fig_notions/escapeThird_zoom.png}}
-    \begin{picture}(1,1)(0,0)
-      {\color{dwhite}{
-        \put(-260,100.5){\circle*{8}}
-        \put(-86,100.5){\circle*{8}}
-        \put(-172,7.5){\circle*{8}}
-      }}
-      \put(-263,98){a}
-      \put(-89,98){b}
-      \put(-175,5){c}
-    \end{picture}
-  \end{tabular}
-  \caption{Aborted detections on side escapes of static directional scans
-           and successful detection using an adaptive directional scan.
-           The last points added to the left of the blurred segment during
-           the initial detection (a) lead to a bad estimation of its
-           orientation, and thus to an incomplete fine detection with
-           a classical directional scanner (b). This scanner is
-           advantageously replaced by an adaptive directional scanner
-           able to continue the segment expansion as far as necessary (c).
-           The input selection is drawn in red color, the scan strip bounds
-           in blue and the detected blurred segment in green.}
-  \label{fig:escape}
-\end{figure}
-
-%\begin{figure}[h]
-%\center
-%  \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.
-%           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.
-%           Here the adaption is visible at the crossing of the tile joins.}
-%  \label{fig:adaption}
-%\end{figure}
-
 \subsection{Control of the assigned width}
 
 The assigned width $\varepsilon$ to the blurred segment recognition algorithm
@@ -278,55 +231,6 @@ In edge selection mode (\RefFig{fig:edgeDir} b), the multi-detection
 algorithm is executed twice, first in main edge selection mode, then
 in opposite edge selection mode.
 
-%Beyond the possible detection of a large set of edges at once, the
-%multi-detection allows the detection of some unaccessible edges in
-%classical single detection mode. This is particularly the case of edges
-%that are quite close to a more salient edge with a higher gradient,
-%as illustrated in \RefFig{fig:voisins}.
-%The multi-detection detects both edges and the user may then select
-%the awaited one.
-%
-%\begin{figure}[h]
-%\center
-%  \begin{tabular}{c@{\hspace{0.2cm}}c@{\hspace{0.2cm}}c@{\hspace{0.2cm}}c}
-%    \includegraphics[width=0.22\textwidth]{Fig_method/voisinImage_zoom.png} &
-%    \includegraphics[width=0.22\textwidth]{Fig_method/voisinGrad_zoom.png} &
-%    \includegraphics[width=0.22\textwidth]{Fig_method/voisinSingle_zoom.png} &
-%    \includegraphics[width=0.22\textwidth]{Fig_method/voisinMulti_zoom.png} \\
-%    \parbox{0.22\textwidth}{\centering{\scriptsize{a)}}} &
-%    \parbox{0.22\textwidth}{\centering{\scriptsize{b)}}} &
-%    \parbox{0.22\textwidth}{\centering{\scriptsize{c)}}} &
-%    \parbox{0.22\textwidth}{\centering{\scriptsize{d)}}}
-%  \end{tabular}
-%  \caption{Detection of close edges with different sharpness:
-%    a) input selection across the edges,
-%    b) gradient map,
-%    c) in single mode, detection of only the edge with the higher gradient,
-%    d) in multi-detection mode, detection of both edges. }
-%  \label{fig:voisins}
-%\end{figure}
-
-%This detection procedure can be used to detect as well straight edges
-%as thin straight objects. In the first case, the gradient vectors of all
-%edge points are assumed to be oriented in the same direction. But if the
-%sign of the gradient direction is not considered, points with gradient in
-%opposite directions are merged to build the same blurred segment, allowing
-%the detection of both edges of a thin linear structure, like for instance
-%the tile joins of \RefFig{fig:edgeDir}.
-
-%On that example, when a straight feature detection is run
-%(\RefFig{fig:edgeDir} a),
-%a thick blurred segment which extends up to four tiles is provided.
-%When a straight edge detection is run, a very thin blurred segment is
-%built to follow only one join edge.
-%The multi-detection can also be applied to both thin object or edge detection.
-%In the latter case, the detection algorithm is run twice using opposite
-%directions, so that in the exemple of figure (\RefFig{fig:edgeDir} b),
-%both edges (in different colors) are highlighted.
-%These two thin blurred segments are much shorter, probably because the
-%tiles are not perfectly aligned.
-%This example illustrates the versatility of the new detector.
-
 \subsection{Automatic blurred segment detection}
 
 An unsupervised mode is also proposed to automatically detect all the
@@ -339,38 +243,6 @@ In the present work, the stroke sweeping step $\delta$ is set to 10 pixels.
 
 \input{Fig_method/algoAuto}
 
-%\RefFig{fig:evalAuto}b gives an idea of the automatic detection performance.
-%In the example of \RefFig{fig:noisy}, hardly perceptible edges are detected
-%despite of a quite textured context.
-%Unsurpringly the length of the detected edges is linked to the initial
-%value of the assigned width, but a large value also augments the rate
-%of interfering outliers insertion.
-%
-%\begin{figure}[h]
-%\center \begin{tabular}{c@{\hspace{0.1cm}}c@{\hspace{0.1cm}}c}
-%    \includegraphics[width=0.32\textwidth]{Fig_method/parpaings.png} &
-%    \includegraphics[width=0.32\textwidth]{Fig_method/parpaings2.png} &
-%    \includegraphics[width=0.32\textwidth]{Fig_method/parpaings3.png}
-%  \end{tabular}
-%  \begin{picture}(1,1)(0,0)
-%    {\color{dwhite}{
-%      \put(-286,-25.5){\circle*{8}}
-%      \put(-171,-25.5){\circle*{8}}
-%      \put(-58,-25.5){\circle*{8}}
-%    }}
-%    \put(-288.5,-28){a}
-%    \put(-173.5,-28){b}
-%    \put(-60.5,-28){c}
-%  \end{picture}
-%  \caption{Automatic detection of blurred segments on a textured image.
-%           a) the input image,
-%           b) automatic detection result with initial assigned width set
-%           to 3 pixels,
-%           c) automatic detection result with initial assigned width set
-%           to 8 pixels.}
-%  \label{fig:noisy}
-%\end{figure}
-
 The automatic detection of blurred segments in a whole image is left
 available for testing in an online demonstration at the following address: \\
 \href{http://ipol-geometry.loria.fr/~kerautre/ipol_demo/AdaptDirBS_IPOLDemo}{

Article/notions.tex

@@ -16,8 +16,7 @@ $0 \leq ax + by - c < \nu$.
 
 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
+$h(\mathcal{L}) = c$ its {\it shift} to origin, and
 $p(\mathcal{L}) = max(|a|,|b|)$ its period (i.e. the length of its
 periodic pattern).
 When $\nu = p(\mathcal{L})$, then $\mathcal{L}$ is the narrowest 8-connected
@@ -25,8 +24,8 @@ 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 $\nu = \varepsilon$.
+of points in $\mathbb{Z}^2$ that all belong to a digital line $\mathcal{L}$
+of arithmetical width $w(\mathcal{L}) = \varepsilon$.
 \end{definition}
 
 A linear-time algorithm to recognize a blurred segment of assigned width
@@ -36,8 +35,10 @@ 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}.
+%It is also the minimal width of the convex hull of $\mathcal{B}$,
+%that can be computed by Melkman's algorithm \cite{Melkman87}.
+The enclosing digital segment $E(\mathcal{B})$ is the section of this
+optimal digital straight line bounded by the end points of $\mathcal{B}$.
 As depicted on \RefFig{fig:bs},
 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
@@ -49,24 +50,28 @@ point $P_i$ is thus controlled by the recognition test $\mu_i < \varepsilon$.
   \caption{A growing blurred segment $\mathcal{B}_i$ :
 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.}
+the assigned width $\varepsilon$, then the new input point is rejected
+and $\mathcal{B}_i = \mathcal{B}_{i-1}$.}
   \label{fig:bs}
 \end{figure}
 
-Associated to this primitive, the following definition of directional scan
-also based on digital straight lines is also used in the following.
+Associated to this primitive, the following definition of a directional scan
+also based on digital straight lines is also used in this work.
 
 \subsection{Directional scan}
 
+\begin{definition}
 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 $\mathcal{D}$.
+\end{definition}
+
 \begin{equation}
 DS = \left\{ S_i = \mathcal{D} \cap \mathcal{N}_i \cap \mathcal{I}
 \left| \begin{array}{l}
 \delta(\mathcal{N}_i) = - \delta^{-1}(\mathcal{D}) \\
-\wedge~ h_0(\mathcal{N}_i) = h_0(\mathcal{N}_{i-1}) + p(\mathcal{D})
+\wedge~ h(\mathcal{N}_i) = h(\mathcal{N}_{i-1}) + p(\mathcal{D})
 \end{array} \right. \right\}
 %S_i = \mathcal{D} \cap \mathcal{N}_i, \mathcal{N}_i \perp \mathcal{D}
 \end{equation}
@@ -104,8 +109,8 @@ At each iteration $i$, the scans $S_i$ and $S_{-i}$ are successively processed.
        \put(-54,32.5){\circle*{14}}
        \put(-161,10.5){\circle*{20}}
      }}
-     \put(-185,112){$A$}
-     \put(-88,14){$B$}
+     \put(-88,13.5){$A$}
+     \put(-185,111.5){$B$}
      \put(-20,98){$\mathcal{D}$}
      \put(-137,64){\color{blue}{$S_0$}}
      \put(-77,94){\color{red}{$S_8$}}
@@ -146,8 +151,10 @@ $\mathcal{N}_i^{A,B}$:
 
 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 $w$. If we note
-$c_3 = x_C\cdot Y_D - y_C\cdot X_D$ and
-$c_4 = X_D\cdot x_C + Y_D\cdot y_C$, it is then defined by
+$c_3 = x_C\cdot Y_D - y_C\cdot X_D$,
+$c_4 = X_D\cdot x_C + Y_D\cdot y_C$,
+$\nu_{\vec{D}} = max (|X_D|,|Y_D|)$,
+it is then defined by
 the following scan strip $\mathcal{D}^{C,\vec{D},w}$ and scan lines
 $\mathcal{N}_i^{C,\vec{D},w}$:
 \begin{equation}
@@ -155,6 +162,6 @@ $\mathcal{N}_i^{C,\vec{D},w}$:
 \mathcal{D}^{C,\vec{D},w}
 = \mathcal{L}(Y_D,~ -X_D,~ c_3 - w / 2,~ w) \\
 \mathcal{N}_i^{C,\vec{D},w} = \mathcal{L}(X_D,~ Y_D,~
-               c_4 - w / 2 + i\cdot w,~ max (|X_D|,|Y_D|)
+               c_4 - w / 2 + i\cdot w,~ \nu_{\vec{D}}
 \end{array} \right.
 \end{equation}

Code/Seg/BlurredSegment/bstracker.cpp

@@ -449,5 +449,3 @@ void BSTracker::switchScanExtent ()
   maxScan = (maxScan == gMap->getHeightWidthMax () ?
              DEFAULT_MAX_SCAN : gMap->getHeightWidthMax ());
 }
-
-