<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\section{Theoretical background}

\label{sec:notions}

\subsection{Blurred segment}

This work relies on the notion of digital straight line as classically
defined in the digital geometry literature \cite{KletteRosenfeld04}.
Only the 2D case is considered here.

\begin{definition}
A digital line $\mathcal{L}(a,b,c,\nu)$, with $(a,b,c,\nu) \in \mathbb{Z}^4$,
is the set of points $P(x,y)$ of $\mathbb{Z}^2$ that satisfy :
$0 \leq ax + by - c &lt; \nu$.
\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
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 $\varepsilon$.
\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.
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 $\mu_{i+1} &lt; \varepsilon$.

\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
the assigned width $\varepsilon$, then the new input point is rejected.}
  \label{fig:bs}
\end{figure}

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 ($\mu_{i+\lambda} = \mu_i$), 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}

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 $\mathcal{D}$.
\begin{equation}
S_i = \mathcal{D} \cap \mathcal{S}_i, \mathcal{S}_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 $\mathcal{S}_0$.

\begin{figure}[h]
\center
  \input{Fig_notions/fig}
  \caption{Example of directional scan.}
  \label{fig:ds}
\end{figure}

A directional scan can be defined by its central 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}
\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 $c_2 = a\cdot x_B + b\cdot y_B$.

The scan line $\mathcal{S}_i$ is then defined by :
\begin{equation}
\mathcal{S}_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.

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 :
\begin{equation}
\mathcal{D}(C,\vec{D},w)
= \mathcal{L}(y_D, -x_D, x_C\cdot y_D - y_C\cdot x_D - w / 2, w)
\end{equation}

\noindent
and the scan line $\mathcal{S}_i(C,\vec{D},w)$ :
\begin{equation}
\mathcal{S}_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
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
on the detected segment.
But only an estimation of this blurred segment direction is provided,
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}
  \caption{Example of early detection failures
           on side escapes from the directional scan.}
  \label{fig:sideEscapes}
\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.
% -&gt; 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.
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.

\begin{figure}[h]
\center
  \begin{picture}(300,40)
  \end{picture}
  \caption{Example of blurred segment detection
           using an adaptive directional scan.}
  \label{fig:adaptiveScan}
\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.
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