Article: notions section enhanced

Article/abstract.tex

   This paper introduces a new straight edge detector in gray-level images
 based on blurred segments, digital objects able to imbed quality measurements
 on the extracted features. This study completes previous works with a better
-estimation of the blurred segment width and orientation, with the help of two
-main improvements : adaptive directional scans and the control of the
-assigned width to the recognition algorithm.
+estimation of the blurred segment width and orientation, with two main
+improvements: adaptive directional scans and the control of the
+assigned width to the detection algorithm.

Article/notions.tex

@@ -20,8 +20,8 @@ $h_P(\mathcal{L}) = c - ax - by$ 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.
+When $\nu = p(\mathcal{L})$, 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
@@ -98,13 +98,14 @@ the scan strip is defined by :
 \mathcal{D}(A,B) = \mathcal{L}(\delta_x, \delta_y, min (c1,c2), 1 + |c_1-c_2|)
 \end{equation}
 \noindent
-where $\delta_x = x_B - x_A, \delta_y = y_B - y_A,
-c_1 = a\cdot x_A + b\cdot y_A$ and $c_2 = a\cdot x_B + b\cdot y_B$.
+where $\delta_x = x_B - x_A$, $\delta_y = y_B - y_A$,
+$c_1 = \delta_x\cdot x_A + \delta_y\cdot y_A$ and
+$c_2 = \delta_x\cdot x_B + \delta_y\cdot y_B$.
 
 The scan line $\mathcal{N}_i$ is then defined by :
 \begin{equation}
 \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})
+\delta_y\cdot x_A - \delta_x\cdot y_A + i\cdot \nu_{AB}, \nu_{AB})
 \end{equation}
 where $\nu_{AB} = max (|\delta_x|, |\delta_y|)$
 
@@ -129,16 +130,17 @@ and the scan line $\mathcal{N}_i(C,\vec{D},w)$ :
 
 \subsection{Adaptive directional scan}
 
-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 (\RefFig{fig:escape}).
-Therefore a second search is run again using a second directional scan aligned
+The blurred segment is searched within a directional scan with a position
+and an orientation approximately provided by the user, or blindly defined
+in unsupervised mode.
+Most of the time, the detection stops where the segment escapes sideways
+from the scan strip (\RefFig{fig:escape}).
+A second search is then run using another directional scan aligned
 on the detected segment.
-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.
+However, even in the case of an exact detection, the estimated orientation
+of the segment 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.
 
 \begin{figure}[h]
 \center
@@ -174,21 +176,23 @@ on a blurred segment escape from the directional scan.
 %  \label{fig:cone}
 %\end{figure}
 
-In the former work, an additional refinement step was run,
-but doing so, the problem was just delayed further.
+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.
-This process could be itered as long as the blurred segment escapes from
+This operation could be itered as long as the blurred segment escapes from
 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.
+But at each iteration, already tested points are processed again,
+thus producing a useless computational cost.
 
-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 $i$.
+Here the proposed solution is to dynamically align the scan direction to
+the blurred segment one all along the expansion stage.
+At each iteration $i+1$, the scan strip is updated using the direction
+of the blurred segment computed at previous iteration $i$.
 The adaptive directional scan $ADS$ is then defined by :
 \begin{equation}
 %S_i = \mathcal{D}_{i-1} \cap \mathcal{N}_i
@@ -197,10 +201,14 @@ 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) \\
 \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)
+\cap \mathcal{D}_{i+1} = D (\mathcal{B}_i,\varepsilon + k)
 \end{array} \right. \right\}
 \end{equation}
-The last clause expresses the update of the scan bounds at step $i+1$.
+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).
+The last clause expresses the update of the scan bounds at iteration $i+1$.
 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}.
@@ -225,8 +233,8 @@ An example of adaptive directional scan is given in \RefFig{fig:adaption}.
 The assigned width $\varepsilon$ to the blurred segment recognition algorithm
 is initially set to a large value $\varepsilon_0$ in order to allow the
 detection of large blurred segments.
-Then, when no more augmentation of the minimal width is observed as the
-blurred segment expends ($\mu_{i+\lambda} = \mu_i$), it is set to a much
+Then, when no more augmentation of the minimal width is observed after
+$\lambda$ iterations ($\mu_{i+\lambda} = \mu_i$), it is set to a much
 stricter value able to circumscribe the possible interpretations of the
 segment, that take into account the digitization margins:
 \begin{equation}